Residue class biases in unrestricted partitions, partitions into distinct parts, and overpartitions

TL;DR

This paper shows that certain number patterns (partitions) have biases in how often specific remainders appear when divided by a fixed number.

Contribution

The paper proves new biases in residue classes for three types of partitions and provides asymptotic formulas for these biases.

Findings

Biases in residue classes for unrestricted partitions are proven.

Asymptotic formulas for symmetric residue class biases are established.

Results apply to partitions into distinct parts and overpartitions.

Abstract

We prove specific biases in the number of occurrences of parts belonging to two different residue classes a and b, modulo a fixed nonnegative integer m, for the sets of unrestricted partitions, partitions into distinct parts, and overpartitions. These biases follow from inequalities for residue-weighted partition functions for the respective sets of partitions. We also establish asymptotic formulas for the numbers of partitions of size n that belong to these sets of partitions and have a symmetric residue class bias (i.e., for \documentclass[12pt]{minimal}
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Full text

Introduction

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {N}}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {N}}_0$$\end{document} denote the sets of positive and nonnegative integers, respectively. Throughout this paper, we assume q to be a fixed complex number satisfying \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<|q|<1$$\end{document} . For any indeterminant a and complex number c, let the q-shifted factorial (cf. [12]) be defined by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (a;q)_\infty :=\prod _{j\ge 0}(1-aq^j),\quad \text {and}\quad (a;q)_c:=\frac{(a;q)_\infty }{(aq^c;q)_\infty }. \end{aligned}$$\end{document}

Products of q-shifted factorials are compactly denoted as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (a_1,\ldots , a_m;q)_c:=\prod _{1\le j\le m}(a_j;q)_c \end{aligned}$$\end{document}

for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\in {\mathbb {N}}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c\in {{\mathbb {C}}}\cup \{\infty \}$$\end{document} .

A partition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} of a positive integer n (cf. [2]) is a nonincreasing sequence of positive integers (called the parts), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda =(\lambda _1, \lambda _2,\ldots ,\lambda _l)$$\end{document} , such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _1+\lambda _2+\cdots +\lambda _l=n$$\end{document} . The integer \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l=:\ell (\lambda )$$\end{document} is called the number of parts of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} . If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} is a partition of n then we write the size of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\lambda |=n$$\end{document} . We also define that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell (\varnothing )=|\varnothing | = 0$$\end{document} for the empty partition, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varnothing $$\end{document} , of 0. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {P}}(n)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {D}}}(n)$$\end{document} denote the sets of all partitions and all partitions with distinct parts (which we shall call “distinct partitions” for short), of n, respectively. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {P}}=\bigcup _{n\ge 0}{\mathcal {P}}(n)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {D}}}=\bigcup _{n\ge 0}{{\mathcal {D}}}(n)$$\end{document} denote the sets of all partitions and distinct partitions, respectively. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a,b, m\in {\mathbb {N}}$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le a<b\le m$$\end{document} . We define \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _{a,m}(\lambda )$$\end{document} to be the number of parts of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} congruent to a modulo m, and define \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _{a,m}(\lambda ,\mu ):=\ell _{a,m}(\lambda )+\ell _{a,m}(\mu )$$\end{document} for each pair \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\lambda , \mu )\in {\mathcal {P}}\times {{\mathcal {D}}}$$\end{document} .

In this paper, we consider the following weighted partition function, defined as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p_n(x,y)=\sum _{\begin{array}{c} |\lambda |+|\mu |= n\\ (\lambda , \mu )\in {\mathcal {P}}\times {{\mathcal {D}}} \end{array}}x^{\ell (\lambda )}y^{\ell (\mu )}. \end{aligned}$$\end{document}

where x and y are nonnegative real numbers, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\in {\mathbb {N}}_0$$\end{document} . It is clear that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(n)=p_n(1,0)$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q(n)=p_n(0,1)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{p}}(n)=p_n(1,1)$$\end{document} are the unrestricted (or ordinary) partition function, distinct partition function, and overpartition function, respectively. (Overpartitions were first explicitly considered by Brenti in [6, Section 3] who called them “dotted partitions.” A thorough independent study of these objects, involving bijections, generating functions and connections to Bailey chains, was initiated by Corteel and Lovejoy [8] who gave them the name “overpartitions,” which we follow.) We will focus on the residue-weighted biases partition functions which we define by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p_{n}(a,b,m; x,y):=\sum _{\begin{array}{c} |\lambda |+|\mu | =n\\ (\lambda , \mu )\in {\mathcal {P}}\times {{\mathcal {D}}}\\ \ell _{a,m}(\lambda ,\mu )> \ell _{b,m}(\lambda ,\mu ) \end{array}} x^{\ell (\lambda )}y^{\ell (\mu )}. \end{aligned}$$\end{document}

From standard arguments of partition theory (cf. [2]), we have the following generating function for the statistics \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell (\lambda )$$\end{document} on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {P}}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell (\mu )$$\end{document} on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {D}}}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _{a,m}(\lambda ,\mu ), \ell _{b,m}(\lambda ,\mu )$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\lambda |+|\mu |$$\end{document} on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {P}}\times {{\mathcal {D}}}$$\end{document} :

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\sum _{(\lambda , \mu )\in {\mathcal {P}}\times {{\mathcal {D}}}} u^{\ell _{a,m}(\lambda ,\mu )}v^{\ell _{b,m}(\lambda ,\mu )} x^{\ell (\lambda )}y^{\ell (\mu )}q^{|\lambda |+|\mu |}\nonumber \\&=\frac{(-yq;q)_\infty ( -uyq^a,-vyq^b,xq^a,xq^b;q^m)_\infty }{(xq;q)_\infty (-yq^a,-yq^b,uxq^a,vxq^b;q^m)_\infty }. \end{aligned}$$\end{document}

Special cases of (1.3) are of interest. The case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x,y)=(1,0)$$\end{document} , that is

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{\lambda \in {\mathcal {P}}}u^{\ell _{a,m}(\lambda )} v^{\ell _{b,m}(\lambda )}q^{|\lambda |}= \frac{(q^a,q^b;q^m)_\infty }{(q;q)_\infty (uq^a,vq^b;q^m)_\infty }, \end{aligned}$$\end{document}

was studied by Chern who in [7, Theorem 1.3] proved that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p_{n}(a,b,m; 1,0)\ge p_{n}(b,a,m; 1,0) \end{aligned}$$\end{document}

holds for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le a<b\le m$$\end{document} . Already earlier, Kim–Kim–Lovejoy [17] proved a phenomenon of parity bias for integer partitions, namely: the number of partitions of n ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n>0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ne 2$$\end{document} ) with more odd parts than even parts is greater than the number of partitions of n ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n>0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ne 2$$\end{document} ) with more even parts than odd parts. In our notation, they proved

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p_{n}(1,2,2; 1,0)\ge p_{n}(2,1,2; 1,0) \end{aligned}$$\end{document}

for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 0$$\end{document} , where the inequality is strict except for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\in \{0, 2\}$$\end{document} . Similar phenomena were shown by Kim–Kim [16] and include the following inequalities:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p_{n}(1,m,m; 1,0)\ge p_{n}(m,1,m; 1,0)\quad \text {and}\quad p_{n}(1,m-1,m; 1,0)\ge p_{n}(m-1,1,m; 1,0), \end{aligned}$$\end{document}

for all integers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\ge 2$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 0$$\end{document} .

While the inequality (1.4) of partitions holds for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 0$$\end{document} , the same inequality holds only from a certain integer on when restricting to the set of distinct partitions. In fact, Kim–Kim–Lovejoy [17] conjectured (in our notation) that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p_{n}(1,2,2; 0,1)> p_{n}(2,1,2; 0,1) \end{aligned}$$\end{document}

for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n>19$$\end{document} (only); this has been proved by Banerjee–Bhattacharjee–Dastidar–Mahanta–Saikia [4] by using a combinatorial approach.

In this paper, we are interested in the biases of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{n}(a,b,m; x,y)$$\end{document} in terms of different residue classes a and b of a fixed modulus m. Our main results are the following three Theorems 1.1–1.3.

Theorem 1.1

Let a, b, m be any integers such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le a<b\le m$$\end{document} . For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\ge 1$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y\ge 0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 0$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p_{n}(a,b,m; x,y)\ge p_{n}(b,a,m; x,y). \end{aligned}$$\end{document}

We point out that the case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x,y)=(1,1)$$\end{document} of Theorem 1.1, that is the case related to the generating function

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{(\lambda , \mu )\in {\mathcal {P}}\times {{\mathcal {D}}}} u^{\ell _{a,m}(\lambda ,\mu )}v^{\ell _{b,m}(\lambda ,\mu )} q^{|\lambda |+|\mu |}= \frac{(-q;q)_\infty (-uq^a,-vq^b,q^a,q^b;q^m)_\infty }{(q;q)_\infty (-q^a,-q^b,uq^a,vq^b;q^m)_\infty }, \end{aligned}$$\end{document}

is the overpartitions analogue of the aforementioned result by Chern [7]. We also establish some new results for the bias of distinct partitions, belonging to the case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x,y)=(0,1)$$\end{document} . In particular, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\ge 0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y=1$$\end{document} , we have the following theorem.

Theorem 1.2

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\ge 0$$\end{document} and a, b, m be integers such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le a<b\le m$$\end{document} . Assume that there exists a positive integer k with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\mid (b-a)$$\end{document} such that neither of the congruences

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} 2^{h}k\equiv a\pmod m\;\;\text {and}\;\; 2^{h}k\equiv b\pmod m, \end{aligned}$$\end{document}

possesses a solution for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h\in {\mathbb {N}}$$\end{document} . Then for all integers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 0$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p_{n}(a,b,m; x,1)\ge p_{n}(b,a,m; x,1). \end{aligned}$$\end{document}

Remark 1.1

Let a, b, m be integers such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le a<b\le m$$\end{document} . It is not difficult to check that the following triples (a, b, m) meet the conditions in Theorem 1.2.

1. Any even integer m and odd integers a, b. (In this case, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=2$$\end{document} guarantees the nonexistence of solutions for any of the two congruences.)
2. Any integers m, a, b such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gcd (a,b, m)$$\end{document} has an odd factor \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ge 3$$\end{document} . (In this case, one can take \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=1$$\end{document} .)

We finally prove the following asymptotic formulas for weighted partitions with biases of symmetric residue classes (by which we mean that in the biases of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_n(a,b,m;x,y)$$\end{document} the two residue classes a and b satisfy the symmetry \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b=m-a$$\end{document} where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le a<m/2$$\end{document} ). Below, when writing a/2m we mean a/(2m), and other fractions in one line notation (with denominators consisting of products) are to be similarly interpreted in the remainder of this paper.

Theorem 1.3

For any given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le a< m/2$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x,y)\in \{(1,0),(0,1),(1,1)\}$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p_{n}(a,m-a,m; x,y)\sim c_{a,m}(x)p_n(x,y), \end{aligned}$$\end{document}

as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\rightarrow +\infty $$\end{document} , where

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left( c_{a,m}(0), c_{a,m}(1)\right) = \left( \frac{1}{2},~\frac{\psi \left( 1/2+a/2m\right) - \psi \left( {a}/{2m}\right) }{2\pi \csc (a\pi /m)}\right) , \end{aligned}$$\end{document}

and where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi (x)=\Gamma '(x)/\Gamma (x)$$\end{document} is the digamma function.

The proof of Theorem 1.3 employs a residue class analogue of Ingham’s Tauberian theorem [14, Theorem 1], which, to the best of the authors’ knowledge, has not appeared before in the literature. We state it as the following theorem and believe that it is of independent interest. We give its proof in Sect. 3.3.

Theorem 1.4

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(q)=\sum _{n\ge 0}c_nq^n$$\end{document} be a power series whose coefficients \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_n$$\end{document} are nonnegative. Suppose that for some positive integer m and any integer \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le h<m$$\end{document} , one has

1. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_{n+m}\ge c_n$$\end{document} for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 0$$\end{document} , and
2. there exist constants \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in {\mathbb {R}}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha , \beta ,\rho \in {\mathbb {R}}_+$$\end{document} such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f(e^{-z})\sim \alpha z^{\gamma }\exp \left( \beta z^{-\rho }/\rho \right) \;\;\text {and}\;\; f(e^{-z}e^{2\pi \textrm{i}h/m})=o\left( f(e^{-z})\right) , \end{aligned}$$\end{document}

when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z\rightarrow 0$$\end{document} in every fixed Stolz angle \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\arg (z)|\le \Delta $$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(0<\Delta <\pi /2)$$\end{document} . Then, as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\rightarrow \infty $$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} c_n\sim \frac{\alpha \beta ^{\frac{1+2\gamma }{2(1+\rho )}}}{\sqrt{2\pi (1+\rho )}}n^{-\frac{1+2\gamma }{2(1+\rho )}-\frac{1}{2}} \exp \left( (1+1/\rho )\beta ^{1/(1+\rho )} n^{\rho /(1+\rho )}\right) . \end{aligned}$$\end{document}

The rest of the paper is organized as follows. In Sect. 2, we give the proof of Theorems 1.1 and 1.2. This is achieved with the help of some q-series transformations and a new overpartitions analogue of a theorem of Andrews [1, Theorem 3]. In Sect. 3, we first provide the proof of our residue class analogue of Ingham’s Tauberian theorem, i.e., of Theorem 1.4. Then, we turn to partition functions with bias of symmetric residue classes and establish their generating functions and asymptotics. We close in Sect. 4 with some remarks on residue class biases in distinct partitions.

The proofs of Theorems 1.1 and 1.2

In this section, we provide proofs of Theorems 1.1 and 1.2. To study \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{n}(a,b,m; x,y)$$\end{document} , we first derive its generating function. Using Cauchy’s q-binomial theorem (cf. [12, Appendix (II.3)]):

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{(\alpha t;q)_\infty }{(t;q)_\infty } =\sum _{n\ge 0}\frac{(\alpha ;q)_nt^n}{(q;q)_n}, \end{aligned}$$\end{document}

valid for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|t|<1$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{(-uyq^a,-vyq^b;q^m)_\infty }{(uxq^a,vxq^b;q^m)_\infty } =\sum _{n_1\ge 0}\frac{(-y/x;q^m)_{n_1}(xuq^{a})^{n_1}}{(q^m;q^m)_{n_1}} \sum _{n\ge 0}\frac{(-y/x;q^m)_{n}(xvq^{b})^{n}}{(q^m;q^m)_{n}}. \end{aligned}$$\end{document}

Therefore, by the definition (1.2) for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{n}(a,b,m; x,y)$$\end{document} and (1.3), we obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{n\ge 0}p_{n}(a,b,m; x,y)q^n&=\frac{(-yq;q)_\infty (xq^a,xq^b;q^m)_\infty }{(xq;q)_\infty (-yq^a,-yq^b;q^m)_\infty }\nonumber \\&\quad \times \sum _{\begin{array}{c} n_1,n\ge 0\\ n_1> n \end{array}} \frac{x^{n_1}(-y/x;q^m)_{n_1}x^{n}(-y/x;q^m)_{n}q^{an_1+bn}}{(q^m;q^m)_{n_1}(q^m;q^m)_{n}}. \end{aligned}$$\end{document}

Writing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_1=k+1+n$$\end{document} in (2.1), and using the simple identity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha ; q)_{n+1+k}=(1-\alpha )(\alpha q;q)_{n+k}$$\end{document} , we get

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (1-q^m)\sum _{n\ge 0}p_{n}(a,b,m; x,y)q^n =\frac{(-yq;q)_\infty (xq^a,xq^b;q^m)_\infty }{(xq;q)_\infty (-yq^a,-yq^b;q^m)_\infty }(x+y)q^a&\nonumber \\ \times \sum _{n,k\ge 0} \frac{(-q^my/x;q^m)_{n+k}(-y/x;q^m)_n x^{2n+k}q^{(a+b)n+ka}}{(q^{2m};q^m)_{n+k}(q^m;q^m)_{n}}&. \end{aligned}$$\end{document}

Equation (2.2) immediately yields the following monotonicity result, which we will use for deriving asymptotic formulas for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{n}(a,m-a,m; x,y)$$\end{document} in Sect. 3.

Lemma 2.1

For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x,y\ge 0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 0$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p_{n+m}(a,b,m; x,y)\ge p_{n}(a,b,m; x,y). \end{aligned}$$\end{document}

Equation (2.2) also implies that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\sum _{n\ge 0} \left( p_{n}(a,b,m; x,y)-p_{n}(b,a,m; x,y)\right) q^n\nonumber \\&=\frac{1}{1-q^m}(1-q^{b-a})\frac{(-yq;q)_\infty (xq^a,xq^b;q^m)_\infty }{(xq;q)_\infty (-yq^a,-yq^b;q^m)_\infty }(x+y)q^a\nonumber \\&\quad \times \sum _{n,k\ge 0}\frac{(1-q^{(k+1)(b-a)})(-q^my/x;q^m)_{n+k} (-y/x;q^m)_n x^{2n+k}q^{(a+b)n+ak}}{(1-q^{b-a})(q^{2m};q^m)_{n+k}(q^m;q^m)_{n}}. \end{aligned}$$\end{document}

Note that the q-series expansion of the above infinite double sum has nonnegative coefficients, with constant term equal to 1, provided that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le a<b\le m$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x,y\ge 0$$\end{document} , because of

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{1-q^{(k+1)(b-a)}}{1-q^{b-a}}=\sum _{0\le j\le k}q^{j(b-a)}. \end{aligned}$$\end{document}

This immediately implies the following lemma.

Lemma 2.2

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le a<b\le m$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x,y\ge 0$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x+y> 0$$\end{document} . Define

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f_{a,b,m,x,y}(q)=(1-q^{b-a})\frac{(-yq;q)_\infty }{(xq;q)_\infty } \frac{(xq^a,xq^b;q^m)_\infty }{(-yq^a,-yq^b;q^m)_\infty }. \end{aligned}$$\end{document}

Then,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p_{n}(a,b,m; x,y)\ge p_{n}(b,a,m; x,y)\ge 0 \end{aligned}$$\end{document}

for all integers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 0$$\end{document} , provided that all the Taylor coefficients of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_{a,b,m,x,y}(q)$$\end{document} about \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q=0$$\end{document} are nonnegative.

Lemma 2.2 will be used to give the proof of Theorem 1.1 in the cases \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(a,b)\ne (1,2)$$\end{document} , and the proof of Theorem 1.2. To achieve that we will need the following proposition.

Proposition 2.3

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\ge 1, y\ge 0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(a,b)\ne (1,2)$$\end{document} , all the Taylor coefficients of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_{a,b,m,x,y}(q)$$\end{document} about q are nonnegative.

Proof

Note that for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<a<b\le m$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f_{a,b,m,x,y}(q)=\frac{1-q^{b-a}}{1-xq}\frac{(-yq;q^m)_\infty }{(xq^{m+1};q^m)_\infty }\prod _{\begin{array}{c} 1\le j\le m\\ j\not \in \{1,a,b\} \end{array}} \frac{(-yq^j;q^m)_\infty }{(xq^j;q^m)_{\infty }} \end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{1-q^{b-a}}{1-x q}= \left( 1+\sum _{j\ge 1}(x^{j}-x^{j-1})q^j\right) \sum _{0\le h<b-a}q^{h}; \end{aligned}$$\end{document}

and for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a=1$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b>2$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f_{a,b,m,x,y}(q)=\frac{1-q^{b-1}}{1-x q^{b-1}} \frac{(-yq^{b-1};q^m)_\infty }{(xq^{m+b-1};q^m)_\infty } \prod _{\begin{array}{c} 1\le j\le m\\ j\not \in \{1,b-1, b\} \end{array}} \frac{(-yq^j;q^m)_\infty }{(xq^j;q^m)_{\infty }} \end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{1-q^{b-1}}{1-x q^{b-1}}=1+\sum _{j\ge 1}(x^{j}-x^{j-1})q^{(b-1)j}. \end{aligned}$$\end{document}

Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\ge 1$$\end{document} , it follows that all the Taylor coefficients of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_{a,b,m,x,y}(q)$$\end{document} about \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q=0$$\end{document} are nonnegative. This completes the proof of the proposition. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

To prove Theorem 1.1 for the remaining case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(a,b)=(1,2)$$\end{document} , we use another method. For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le a<b\le m$$\end{document} , by writing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_1=k+1+n$$\end{document} in (2.1) and using the elementary identity

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (\alpha ; q)_{n+1+k}=(\alpha ;q)_n(\alpha q^{n};q)_{k+1}, \end{aligned}$$\end{document}

we find that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{n\ge 0}p_{n}(a,b,m; x,y)q^n&= \frac{(-yq;q)_\infty (xq^a,xq^b;q^m)_\infty }{(xq;q)_\infty (-yq^a,-yq^b;q^m)_\infty }\\&\quad \times \sum _{n\ge 0}\frac{(-y/x;q^m)_{n}^2 (x+yq^{mn})x^{2n}q^{(a+b)n}}{(q^m;q^m)_{n}^2}\\&\quad \times \sum _{k\ge 0}\frac{(-q^{m(n+1)}{y}/{x};q^m)_{k} x^kq^{a(k+1)}}{(q^{m(n+1)};q^m)_{k+1}}. \end{aligned}$$\end{document}

Thus,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\sum _{n\ge 0}\left( p_{n}(a,b,m; x,y)-p_{n}(b,a,m; x,y)\right) q^n\nonumber \\&=\frac{(-yq;q)_\infty (xq^a,xq^b;q^m)_\infty }{(xq;q)_\infty (-yq^a,-yq^b;q^m)_\infty } \sum _{n\ge 0}\frac{(-y/x;q^m)_{n}^2 (x+yq^{mn})x^{2n}q^{(a+b)n}}{(q^m;q^m)_{n}^2}\nonumber \\&\quad \times \left( \sum _{k\ge 0}\frac{(-q^{m(n+1)}{y}/{x};q^m)_{k} x^kq^{a(k+1)}}{(q^{m(n+1)};q^m)_{k+1}}- \sum _{k\ge 0}\frac{(-q^{m(n+1)}{y}/{x};q^m)_{k}x^kq^{b(k+1)}}{(q^{m(n+1)};q^m)_{k+1}}\right) . \end{aligned}$$\end{document}

From Equation (2.3), it is obvious that to prove \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{n}(1,2,m; x,y)\ge p_{n}(2,1,m; x,y)$$\end{document} , it suffices to prove the following theorem, which restricts to such pairs (a, b) where b is a multiple of a (and thus can be replaced by ab), which is more than we need for the case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(a,b)=(1,2)$$\end{document} .

Theorem 2.4

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\ge 1$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y\ge 0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a,b\in {\mathbb {N}}$$\end{document} . For all integers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m,s\ge 1$$\end{document} , the q-series expansion of

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{k\ge 0}\frac{(-yq^{s}/x;q^m)_{k}x^{k}q^{a(k+1)}}{(q^{s};q^m)_{k+1}} -\sum _{k\ge 0}\frac{(-yq^{s}/x;q^m)_{k}x^{k}q^{ab(k+1)}}{(q^{s};q^m)_{k+1}} \end{aligned}$$\end{document}

has nonnegative coefficients.

We note that the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x,y,a,b)=(1,0,1,2)$$\end{document} special case of Theorem 2.4 was conjectured by Chern [7, Conjecture 4.2] and subsequently proved by Binner [5, Section 2], which we state as the following corollary:

Corollary 2.5

For all integers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m,s\ge 1$$\end{document} , the q-series expansion of

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{k\ge 0}\frac{q^k(1-q^k)}{(q^{s};q^m)_k} \end{aligned}$$\end{document}

has nonnegative coefficients.

Proof of Theorem 2.4

Using the Heine transformation [12, Appendix (III.2)] (valid for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|z|<1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\gamma /\beta |<1$$\end{document} )

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{n\ge 0}\frac{(\alpha ,\beta ;q)_n}{(\gamma ,q;q)_n}z^n= \frac{(\gamma /\beta ,\beta z;q)_\infty }{(\gamma ,z;q)_\infty } \sum _{n\ge 0}\frac{(\alpha \beta z/\gamma ,\beta ;q)_n}{(\beta z,q;q)_n}(\gamma /\beta )^n, \end{aligned}$$\end{document}

and making the substitution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \mapsto q, \gamma \mapsto \gamma q$$\end{document} , one readily obtains

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{n\ge 0}\frac{(\alpha ;q)_n}{(\gamma ;q)_{n+1}}z^n= \sum _{n\ge 0}\frac{(\alpha z/\gamma ;q)_n}{(z;q)_{n+1}}\gamma ^n \end{aligned}$$\end{document}

(which can be referred to as Fine’s transformation [10, Equation (6.3)], as Fine thoroughly studied this type of series in his monograph on basic hypergeometric series), valid for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|z|<1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\gamma |<1$$\end{document} . Thus,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\sum _{k\ge 0}\frac{(-yq^{s}/x;q^m)_{k}(q^a(xq^{a})^k- q^{ab}(xq^{ab})^k)}{(q^{s};q^m)_{k+1}}\\ &= \sum _{k\ge 0}\left( \frac{q^a(-yq^a;q^m)_{k}}{(xq^{a};q^m)_{k+1}}- \frac{q^{ab}(-yq^{ab};q^m)_{k}}{(xq^{ab};q^m)_{k+1}}\right) q^{sk}. \end{aligned}$$\end{document}

Note that, for any integer \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k \ge 0$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\frac{(-yq^a;q^m)_{k}xq^a}{(xq^a;q^m)_{k+1}}- \frac{(-yq^{ab};q^m)_{k}xq^{ab}}{(xq^{ab};q^m)_{k+1}}\\ &= \sum _{h\ge 1}\left( \frac{(-yq^a;q^m)_{k}(xq^a)^{h}}{(xq^a;q^m)_{k}} -\frac{(-yq^{ab};q^m)_{k}(xq^{ab})^{h}}{(xq^{ab};q^m)_{k}}\right) q^{(h-1)km}. \end{aligned}$$\end{document}

The proof of Theorem 2.4 now is an immediate consequence of Theorem 2.6, which we provide subsequently. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

The following result is an overpartition analogue of a result by Andrews [1, Theorem 3].

Theorem 2.6

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(a_j)_{j\ge 0},(b_j)_{j\ge 0}$$\end{document} be two increasing sequences of positive integers with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_0\equiv 0 \pmod {a_0}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_0>a_0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_j-a_j\equiv 0\pmod {a_0}$$\end{document} for each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j\ge 1$$\end{document} . For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\in {\mathbb {N}}_0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h\in {\mathbb {N}}_0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\ge 1, y\ge 0$$\end{document} , the q-series expansion of

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (xq^{a_0})^h\prod _{0\le j\le n}\frac{1+yq^{a_j}}{1-xq^{a_j}} -(xq^{b_0})^h\prod _{0\le j\le n}\frac{1+yq^{b_j}}{1-xq^{b_j}} \end{aligned}$$\end{document}

has nonnegative coefficients.

Proof

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=0$$\end{document} , the q-series expansion of

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\frac{1+yq^{a_0}}{1-xq^{a_0}} (xq^{a_0})^h- \frac{1+yq^{b_0}}{1-xq^{b_0}} (xq^{b_0})^h\\&=(xq^{a_0})^h-(xq^{b_0})^h+(1+y/x) \sum _{k>h}((xq^{a_0})^{k}-(xq^{b_0})^k)\\&=x^hq^{a_0h}+(1+y/x)\sum _{h<k< b_0h/a_0}x^{k}q^{a_0k} +yx^{h-1}q^{b_0h}\\&\quad +(1+y/x)\sum _{\begin{array}{c} k\ge h\\ b_0\not \mid a_0k \end{array}}x^kq^{a_0k} +(1+y/x)\sum _{k\ge h}(x^{b_0k/a_0}-x^{k})q^{b_0k} \end{aligned}$$\end{document}

has nonnegative coefficients (since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\ge 1$$\end{document} ). Note that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&(xq^{a_0})^h\prod _{0\le j\le n}\frac{1+yq^{a_j}}{1-xq^{a_j}}-(xq^{b_0})^h\prod _{0\le j\le n} \frac{1+yq^{b_j}}{1-xq^{b_j}}\nonumber \\&=\left( \frac{1+y q^{a_n}}{1-xq^{a_n}}- \frac{1+y q^{b_n}}{1-xq^{b_n}}\right) (xq^{a_0})^h \prod _{0\le j< n}\frac{1+yq^{a_j}}{1-xq^{a_j}}\nonumber \\&\quad +\frac{1+y q^{b_n}}{1-xq^{b_n}} \left( (xq^{a_0})^h\prod _{0\le j\le n-1} \frac{1+yq^{a_j}}{1-xq^{a_j}} -(xq^{b_0})^h\prod _{0\le j\le n-1} \frac{1+yq^{b_j}}{1-xq^{b_j}}\right) , \end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\left( \frac{1+y q^{a_n}}{1-xq^{a_n}}-\frac{1+y q^{b_n}}{1-xq^{b_n}}\right) \prod _{0\le j< n}\frac{1+yq^{a_j}}{1-xq^{a_j}}\nonumber \\&=\frac{ 1-q^{b_n-a_n}}{1-xq^{a_0}}\frac{q^{a_n}(x+y)}{1-x q^{b_n}} \prod _{0\le j< n}\frac{1+yq^{a_j}}{1-xq^{a_{j+1}}}. \end{aligned}$$\end{document}

Now, since the q-series expansion of the factor

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{1-q^{b_n-a_n}}{1-xq^{a_0}}= \left( 1+\sum _{j\ge 1}(x^{j}-x^{j-1})q^{a_0j}\right) \sum _{0\le j< (b_n-a_n)/a_0}q^{a_0 j} \end{aligned}$$\end{document}

has nonnegative coefficients, we can conclude that the q-series expansion of (2.5) has nonnegative coefficients as well. Using (2.4), the proof is established by induction. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

We are now ready to give the proof of Theorem 1.2.

Proof of Theorem 1.2

Since both of the congruence equations

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} 2^{h+1}k\equiv a\pmod m\;\;\text {and}\;\; 2^{h+1}k\equiv b\pmod m \end{aligned}$$\end{document}

possess no solution for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h\in {\mathbb {N}}_0$$\end{document} , we have that both of the equations

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} 2^{h_1}k=m\ell _1+a\;\; \text {and}\;\; 2^{h_2}k=m\ell _2+b \end{aligned}$$\end{document}

possess no nonnegative integer solutions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(h_1,\ell _1)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(h_2, \ell _2)$$\end{document} . This yields

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \{2^hk: h\in {\mathbb {N}}_0\}\subseteq \{j\in {\mathbb {N}}: j\not \equiv a, b\pmod m\}. \end{aligned}$$\end{document}

On the other hand, since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b>a$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\mid (b-a)$$\end{document} , it is clear that the q-series expansion of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1-q^{b-a})/(1-q^{k})$$\end{document} has nonnegative coefficients. Thus,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f_{a,b,m,x,1}(q)=\frac{(1-q^{k})(-q;q)_\infty }{(-q^a,-q^{b};q^m)_\infty }\cdot \frac{1-q^{b-a}}{1-q^k} \prod _{\begin{array}{c} 1\le j\le m\\ j\not \in \{a,b\} \end{array}}\frac{1}{(xq^j;q^m)_{\infty }} \end{aligned}$$\end{document}

has nonnegative coefficients in its q-expansion too, since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\ge 0$$\end{document} and the q-series expansion of

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{(1-q^{k})(-q;q)_\infty }{(-q^a,-q^{b};q^m)_\infty } =(1-q^{k})\prod _{h\ge 0}(1+q^{2^hk}) \prod _{\begin{array}{c} j\ge 1\\ j\not \equiv a,b\pmod m\\ j\ne 2^{h}k, h\in {\mathbb {N}}_0 \end{array}} \left( 1+q^j\right)&\\= \prod _{\begin{array}{c} j\ge 1\\ j\not \equiv a,b\pmod m\\ j\ne 2^{h}k, h\in {\mathbb {N}}_0 \end{array}} \left( 1+q^j\right)&\end{aligned}$$\end{document}

has nonnegative coefficients. Therefore, by using Lemma 2.2 and Proposition 2.3 we get

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p_{n}(a,b,m;x,1)-p_{n}(b,a,m;x,1)\ge 0, \end{aligned}$$\end{document}

which completes the proof of Theorem 1.2. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Weighted partitions with bias of symmetric residue classes

Recall that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_n(a,b,m;x,y)$$\end{document} is defined by (1.2). Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x,y)\in S$$\end{document} where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S=\{(0,1), (1,0), (1,1)\}$$\end{document} , and let a, m be positive integers such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le a<m$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\ne 2a$$\end{document} . We write

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} G_{xy}(a,m;q)=\sum _{n\ge 0}p_n(a,m-a,m;x,y)q^n \end{aligned}$$\end{document}

for the generating function of weighted partitions with bias of symmetric residue classes. The first objective of this section, rather than just using (2.1), is to derive forms for the generating functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{xy}(a,m;q)$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x,y)\in S$$\end{document} , which will be convenient for establishing asymptotic formulas for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_n(a,m-a,m;x,y)$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x,y)\in S$$\end{document} .

Generating functions for weighted partitions with bias of symmetric residue classes

To obtain the generating functions for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{xy}(a,m;q)$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x,y) \in S$$\end{document} , we require the following identities:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (-\zeta , -q/\zeta , q;q)_\infty = \sum _{n\in {\mathbb {Z}}}q^{\frac{n(n-1)}{2}}\zeta ^n, \end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{(q;q)_\infty ^2}{(\zeta , q/\zeta ;q)_\infty }= \sum _{n\in {\mathbb {Z}}} \frac{(-1)^nq^{\frac{n(n+1)}{2}}}{1-\zeta q^n}, \end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{(-q/\zeta ,-\zeta ;q)_\infty }{(q/\zeta ,\zeta ;q)_\infty }= \frac{(-q;q)_\infty ^2}{(q;q)_\infty ^2}\left( 1+2\sum _{n\ge 1} \frac{\zeta ^n+(q/\zeta )^n}{1+q^n}\right) . \end{aligned}$$\end{document}

The identity (3.1) is the well-known Jacobi triple product identity (cf. [12, Appendix (II.28)]). The identity (3.2) is the reciprocal of a theta function, which is expanded in terms of partial fractions by the Mittag–Leffler theorem. (For details, see Ramanujan’s lost notebook [3, Entry 3.2.1] or Garvan [11, Equation (7.15)].) The identity (3.3) is a special case of an identity originally due to Kronecker (cf. [18, pp. 70–71]) and can be easily derived from Ramanujan’s \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_1\psi _1$$\end{document} summation formula (cf. [12, Appendix (II.29)]).

Proof of identity (3.3)

Ramanujan’s \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{1}\psi _{1}$$\end{document} summation formula is

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{n\ge 0}\frac{(a;q)_n}{(b;q)_n}\zeta ^n+ \sum _{n\ge 1}\frac{(q/b;q)_{n}}{(q/a;q)_n}(b/a\zeta )^{n}= \frac{(b/a,q/a\zeta ,a\zeta ,q;q)_\infty }{(b,b/a\zeta ,q/a,\zeta ;q)_\infty }, \end{aligned}$$\end{document}

provided \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\zeta |<1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|b/a\zeta |<1$$\end{document} . Letting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b\mapsto aq$$\end{document} , this reduces to Kronecker’s bilateral summation

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{n\ge 0}\frac{1-a}{1-aq^n}\zeta ^n+ \sum _{n\ge 1}\frac{1-1/a}{1-q^n/a}(q/\zeta )^{n}= \frac{(q,q/a\zeta ,a\zeta ,q;q)_\infty }{(aq,q/\zeta ,q/a,\zeta ;q)_\infty }. \end{aligned}$$\end{document}

Letting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a\rightarrow -1$$\end{document} , we obtain after some rewriting (3.3). \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

We now give the generating functions for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{xy}(a,m;q)$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x,y) \in S$$\end{document} .

Lemma 3.1

We have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} G_{01}(a,m;q)=\frac{(q^2;q^2)_\infty }{(-q^a,-q^{m-a},q^m;q^m)_\infty (q;q)_\infty } \sum _{n\ge 1}q^{\frac{mn(n-1)}{2}+n a}, \\ G_{10}(a,m;q)=\frac{(q^a,q^{m-a};q^m)_\infty }{(q;q)_\infty (q^m;q^m)_\infty ^2}\sum _{n\ge 0}(-1)^{n} \frac{q^{m\frac{n(n+1)}{2}+mn+a}}{1-q^{mn+a}}, \end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} G_{11}(a,m;q)=2\frac{(q^2;q^2)_\infty (q^{2m};q^{2m})_\infty ^2}{(q;q)_\infty ^2(q^m;q^m)_\infty ^4}\frac{(q^a,q^{m-a};q^m)_\infty }{(-q^a,-q^{m-a};q^m)_\infty }\sum _{n\ge 1}\frac{q^{an}}{1+q^{mn}}. \end{aligned}$$\end{document}

Proof

Recall that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} G_{xy}(a,m;q)=\sum _{\begin{array}{c} (\lambda , \mu )\in {\mathcal {P}}\times {{\mathcal {D}}}\\ \ell _{a,m}(\lambda ,\mu )-\ell _{b,m}(\lambda ,\mu )>0 \end{array}} x^{\ell (\lambda )}y^{\ell (\mu )}q^{|\lambda |+|\mu |}, \end{aligned}$$\end{document}

and, by the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u=t$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v=t^{-1}$$\end{document} case of (1.3),

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\sum _{(\lambda , \mu )\in {\mathcal {P}}\times {{\mathcal {D}}}} t^{\ell _{a,m}(\lambda ,\mu )-\ell _{b,m}(\lambda ,\mu )} x^{\ell (\lambda )}y^{\ell (\mu )}q^{|\lambda |+|\mu |}\\ &= \frac{(-yq;q)_\infty ( xq^a,xq^b;q^m)_\infty ( -tyq^a,-t^{-1}yq^b;q^m)_\infty }{(xq;q)_\infty (-yq^a,-yq^b;q^m)_\infty (txq^a,t^{-1}xq^b;q^m)_\infty }. \end{aligned}$$\end{document}

Using identities (3.1)–(3.3), we have:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{\mu \in {{\mathcal {D}}}}t^{\ell _{a,m}(\mu )-\ell _{m-a,m}(\mu )}q^{|\mu |}= \frac{(-q;q)_\infty }{(-q^a,-q^{m-a},q^m;q^m)_\infty } \sum _{n\in {\mathbb {Z}}}q^{\frac{mn(n-1)}{2}+n a}t^{n}, \end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{\mu \in {\mathcal {P}}}t^{\ell _{a,m}(\mu )-\ell _{m-a,m}(\mu )}q^{|\mu |}= \frac{(q^a,q^{m-a};q^m)_\infty }{(q;q)_\infty (q^m;q^m)_\infty ^2} \sum _{n\in {\mathbb {Z}}}\frac{(-1)^{n}q^{m\frac{n(n+1)}{2}}}{1-q^{mn+a}t}, \end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{(\lambda , \mu )\in {\mathcal {P}}\times {{\mathcal {D}}}} t^{\ell _{a,m}(\lambda ,\mu )-\ell _{m-a,m}(\lambda ,\mu )}q^{|\lambda |+|\mu |}&=\frac{(q^2;q^2)_\infty (q^{2m};q^{2m})_\infty ^2(q^a,q^{m-a};q^m)_\infty }{(q;q)_\infty ^2(q^m;q^m)_\infty ^4(-q^a,-q^{m-a};q^m)_\infty }\nonumber \\&\quad \times \left( 1+2\sum _{n\ge 1}\frac{q^{an}t^{n}+q^{(m-a)n}t^{-n}}{1+q^{mn}}\right) . \end{aligned}$$\end{document}

Therefore, by collecting the coefficients of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t^n$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \in {\mathbb {N}}$$\end{document} (and omitting the other coefficients where n is negative) in the above equations (3.4)–(3.6), we readily obtain the proof of the lemma. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Asymptotics of the generating functions

In this subsection, we study the asymptotics of the q-series \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{xy}(a,m;q)$$\end{document} , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x,y)\in S$$\end{document} , q nearing the m-th root of unity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\zeta _{m}^h:=e^{2\pi \textrm{i}h/m}$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\in {\mathbb {N}}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h\in {\mathbb {N}}_0$$\end{document} . In order to achieve this, we first need the following proposition which are special cases of results by Katsurada [15, Theorem 5, Corollary 1.1, Corollary 1.3].

Proposition 3.2

The following asymptotic formulas hold for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z\rightarrow 0$$\end{document} in every fixed Stolz angle \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\arg (z)|\le \Delta $$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(0<\Delta <\pi /2)$$\end{document} .

1. For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h\in {\mathbb {N}}_0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\in {\mathbb {N}}$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gcd (h,m)=1$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (e^{-z}\zeta _{m}^h;e^{-z}\zeta _{m}^h)_\infty = A_0(h,m)z^{-1/2}\exp \left( -\frac{\pi ^2}{6m^2z}\right) (1+O(z)), \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_0(h,m)$$\end{document} is a constant. In particular, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_0(1,1)=\sqrt{2\pi }$$\end{document} . 2. For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha >0$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (-e^{-\alpha z};e^{-z})_\infty = 2^{1/2-\alpha }\exp \left( \frac{\pi ^2}{12z}\right) (1+O(z)). \end{aligned}$$\end{document}

3. For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha , \mu \in (0,1)$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (e^{-\alpha z}, e^{-(1-\alpha ) z};e^{-z})_{\infty }= 2\sin (\pi \alpha )\exp \left( -\frac{\pi ^2}{3z}\right) (1+O(z)), \end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&(e^{2\pi \textrm{i}\mu }e^{-\alpha z},e^{2\pi \textrm{i}(1-\mu )}e^{-(1-\alpha ) z};e^{-z})_{\infty }\\&=e^{-2\pi \textrm{i}(1/2-\alpha )(1/2-\mu )}\exp \left( -\frac{2\pi ^2B_2(\mu )}{z}\right) (1+O(z)), \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_2(\mu )=\mu ^2-\mu +1/6$$\end{document} ,1

The asymptotics of the infinite products in Lemma 3.1 will follow from Proposition 3.2. It remains to establish the asymptotics for the infinite sums in Lemma 3.1. In the following, let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {1}}_{event}$$\end{document} denote the indicator function, let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h\in {\mathbb {N}}_0$$\end{document} and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q=e^{-z/m}\zeta _m^h$$\end{document} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi (u)=\Gamma '(u)/\Gamma (u)$$\end{document} be the digamma function given by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \psi (u)=-\gamma +\sum _{n\ge 1}\left( \frac{1}{n}-\frac{1}{n+u}\right) . \end{aligned}$$\end{document}

We prove the following lemmas.

Lemma 3.3

As \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z\rightarrow 0$$\end{document} in every fixed Stolz angle \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\arg (z)|\le \Delta $$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(0<\Delta <\pi /2)$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{n\ge 1}q^{m\frac{n(n-1)}{2}+n a}= \frac{{\textbf {1}}_{m\mid ah}}{2}\left( \frac{2\pi }{z}\right) ^{1/2}+O(1), \end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{n\ge 0}(-1)^{n}\frac{q^{m\frac{n(n+1)}{2}+mn+a}}{1-q^{mn+a}}=\frac{{\textbf {1}}_{m\mid ah}}{2z} \left( \psi \left( \frac{m+a}{2m}\right) - \psi \left( \frac{a}{2m}\right) \right) +O(|z|^{-1/2}), \end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{n\ge 1}\frac{q^{an}}{1+q^{mn}}= \frac{{\textbf {1}}_{m\mid ah}}{2z}\left( \psi \left( \frac{m+a}{2m}\right) - \psi \left( \frac{a}{2m}\right) \right) +O(1). \end{aligned}$$\end{document}

Proof

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z=x+\textrm{i}y$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x>0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y\in {\mathbb {R}}$$\end{document} , then we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|y|\le x\cdot \tan (\Delta )$$\end{document} . The proofs of (3.7) and (3.9) use similar arguments. In fact, note that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} S(u):=\sum _{1\le n\le u}\zeta _{m}^{ahn}=u\cdot {\textbf {1}}_{m\mid ah}+O(1), \end{aligned}$$\end{document}

uniformly for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u>0$$\end{document} . Hence, Abel’s summation formula implies

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{n\ge 1} &e^{-z\frac{n(n-1)}{2}-\frac{an}{m}z}\zeta _{m}^{ahn} =S(u)e^{-\frac{u(u-1)}{2}z-\frac{a}{m}uz}\big |_{u=0}^{+\infty }-\int \limits _{0}^{\infty }S(u)\left( e^{-\frac{u(u-1)}{2}z-\frac{a}{m}uz}\right) '\textrm{d}u\\&={\textbf {1}}_{m\mid ah}\cdot \int \limits _{0}^{\infty }u \left( e^{-\frac{u(u-1)}{2}z-\frac{a}{m}uz}\right) '\textrm{d}u+ O\left( \int \limits _{0}^{\infty } \left| \left( e^{-\frac{u(u-1)}{2}z-\frac{a}{m}uz}\right) '\right| \textrm{d}u\right) \\&={\textbf {1}}_{m\mid ah}\cdot \int \limits _{0}^{\infty }e^{-\frac{u(u-1)}{2}z-\frac{a}{m}uz}\textrm{d}u+ O\left( |z|\int \limits _{0}^{\infty }\left| u-\frac{1}{2}+ \frac{a}{m}\right| e^{-\frac{u(u-1)}{2}x-\frac{a}{m}ux}\textrm{d}x\right) \\&={\textbf {1}}_{m\mid ah}\cdot \int \limits _{\frac{a}{m}-\frac{1}{2}}^{\infty } e^{-\frac{u^2z}{2}+\frac{(a/m-1/2)^2}{2}z}\textrm{d}u+ O\left( |z|\int \limits _{\frac{a}{m}-\frac{1}{2}}^{\infty }|u| e^{-\frac{u^2x}{2}+\frac{(a/m-1/2)^2}{2}x}\textrm{d}u\right) \\&=\frac{{\textbf {1}}_{m\mid ah}}{2}\left( \frac{2\pi }{z}\right) ^{1/2}+O(1). \end{aligned}$$\end{document}

Abel’s summation formula further implies

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{n\ge 1} &\frac{e^{-azn/m}}{1+e^{-nz}}e^{2\pi \textrm{i}ahn/m} =S(u)\frac{e^{-azu/m}}{1+e^{-uz}}\bigg |_{u=0}^{+\infty }- \int \limits _{0}^{\infty }S(u)\left( \frac{e^{-azu/m}}{1+e^{-uz}}\right) '\textrm{d}u\\&={\textbf {1}}_{m\mid ah}\cdot \int \limits _{0}^{\infty }u \left( \frac{e^{-azu/m}}{1+e^{-uz}}\right) '\textrm{d}u+ O\left( \int \limits _{0}^{\infty }\left| \left( \frac{e^{-azu/m}}{1+e^{-uz}}\right) '\right| \textrm{d}u\right) \\&={\textbf {1}}_{m\mid ah}\cdot \int \limits _{0}^{\infty }\frac{e^{-azu/m}}{1+e^{-uz}}\textrm{d}u+O\left( |z|\int \limits _{0}^{\infty } \frac{e^{aux/m}+e^{-(1-a/m)ux}}{|e^{auz/m}+e^{-(1-a/m)uz}|^2}\textrm{d}x\right) \\&=\frac{{\textbf {1}}_{m\mid ah}}{z}\int \limits _{0}^{\infty } \sum _{k\ge 0}(-1)^{k}e^{-(k+a/m)u}\textrm{d}u+ O\left( 1+\int \limits _{1/|z|}^{\infty }\frac{|z|e^{-aux/m}\textrm{d}u}{(1-e^{-ux})^2}\right) \\&=\frac{{\textbf {1}}_{m\mid ah}}{z}\sum _{k\ge 0}\frac{(-1)^{k}}{k+a/m}+O(1). \end{aligned}$$\end{document}

By using [9, Eq. 5.7.7], we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{k\ge 0}\frac{(-1)^k}{k+a/m}= \frac{1}{2}\left( \psi \left( \frac{m+a}{2m}\right) - \psi \left( \frac{a}{2m}\right) \right) , \end{aligned}$$\end{document}

which completes the proof of (3.9).

It remains to give the proof of (3.8). Note that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\sum _{n\ge 0}(-1)^{n}\frac{q^{m\frac{n(n+1)}{2}+mn+a}}{1-q^{mn+a}}\\ &=\sum _{k\ge 0}\left( \frac{(1-q^m)q^{mk(2k+1)-a}}{(q^{-(2mk+a)}-1)(q^{-(2km+m+a)}-1)}+ \frac{(1-q^{m(2k+1)})q^{mk(2k+1)}}{q^{-(2mk+a)}-1}\right) . \end{aligned}$$\end{document}

We have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{k\ge 0}\frac{(1-q^{m(2k+1)})q^{mk(2k+1)}}{q^{-(2mk+a)}-1}&\ll \sum _{k\ge 0}\frac{|1-e^{-(2k+1)z}|e^{-k^2x}}{e^{(2k+a/m)x}-1}\\&\ll \sum _{0\le k\le 1/|z|}\frac{(2k+1)|z|e^{-k^2x}}{(2k+a/m)x}+ \sum _{k\ge 1/|z|}e^{-k^2x}\\&\ll \sum _{k\ge 0}e^{-k^2x}\ll x^{-1/2}\ll |z|^{-1/2}, \end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\sum _{k\ge 0}\frac{(1-q^m)q^{mk(2k+1)-a}}{(q^{-(2mk+a)}-1)(q^{-(2km+m+a)}-1)}\\&= \sum _{0\le k\le |z|^{-1/2}}\frac{(1-e^{-z})e^{-z(k(2k+1)-a/m)}\zeta _m^{-ah}}{(e^{(2k+a/m)z}\zeta _{m}^{ah}-1)(e^{(2k+1+a/m)z}\zeta _{m}^{ah}-1)}\\&\qquad +O\left( \sum _{k\ge |z|^{-1/2}}\frac{|z|e^{-k^2x}}{(2k+a/m)(2k+1+a/m)x^2}\right) . \end{aligned}$$\end{document}

For the O-term, we estimate that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{k\ge |z|^{-1/2}}\frac{|z|e^{-k^2x}}{(2k+a/m)(2k+1+a/m)x^2}\ll \frac{1}{|z|} \sum _{k\ge |z|^{-1/2}}\frac{1}{k^2}\ll |z|^{-1+1/2}\ll |z|^{-1/2}. \end{aligned}$$\end{document}

We now estimate the main term. Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\le k\le |z|^{-1/2}$$\end{document} , for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\not \mid ah$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{0\le k\le |z|^{-1/2}}\frac{(1-e^{-z})e^{-z(k(2k+1)-a/m)}\zeta _m^{-ah}}{(e^{(2k+a/m)z}\zeta _{m}^{ah}-1)(e^{(2k+1+a/m)z}\zeta _{m}^{ah}-1)}&\\\ll \sum _{0\le k\le |z|^{-1/2}}\frac{|z|}{\left| |\zeta _{m}^{ah}-1|+ O(z^{1/2})\right| ^2}\ll |z|^{1/2}&. \end{aligned}$$\end{document}

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\mid ah$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\sum _{0\le k\le |z|^{-1/2}}\frac{(1-e^{-z})e^{-z(k(2k+1)-a/m)}\zeta _m^{-ah}}{(e^{(2k+a/m)z}\zeta _{m}^{ah}-1)(e^{(2k+1+a/m)z}\zeta _{m}^{ah}-1)}\\&=\sum _{0\le k\le |z|^{-1/2}}\frac{z}{(2k+a/m)(2k+1+a/m)z^2} (1+O((k+1)^2z))(1+O((k+1)z))\\&=\frac{1}{z}\sum _{0\le k\le |z|^{-1/2}}\frac{1}{(2k+a/m)(2k+1+a/m)}+ O\left( \sum _{0\le k\le |z|^{-1/2}}1\right) \\&=\frac{1}{z}\sum _{k\ge 0} \frac{(-1)^k}{k+a/m}+O(|z|^{-1/2}). \end{aligned}$$\end{document}

Combining the above estimates, we get

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{n\ge 0}(-1)^{n}\frac{q^{m\frac{n(n+1)}{2}+mn+a}}{1-q^{mn+a}}= \frac{{\textbf {1}}_{m\mid ah}}{2z}\left( \psi \left( \frac{m+a}{2m}\right) - \psi \left( \frac{a}{2m}\right) \right) +O(|z|^{-1/2}), \end{aligned}$$\end{document}

which completes the proof of the lemma. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Lemma 3.4

We have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} G_{01}\!\left( a,m;e^{-z/m}\right) = \frac{1+O(|z|^{1/2})}{2} \frac{1}{\sqrt{2}}\exp \left( \frac{\pi ^2m}{12z}\right) , \\ G_{10}\!\left( a,m;e^{-z/m}\right) =(1+O(|z|^{1/2})) \frac{\psi \left( \frac{m+a}{2m}\right) - \psi \left( \frac{a}{2m}\right) }{2\pi \csc (a\pi /m)} \left( \frac{z}{2\pi m}\right) ^{1/2}\exp \left( \frac{\pi ^2m}{6z}\right) , \end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} G_{11}\!\left( a,m;e^{-z/m}\right) =(1+O(|z|^{1/2})) \frac{\psi \left( \frac{m+a}{2m}\right) - \psi \left( \frac{a}{2m}\right) }{2\pi \csc (a\pi /m)} \left( \frac{z}{4\pi m}\right) ^{1/2}\exp \left( \frac{\pi ^2m}{4z}\right) \end{aligned}$$\end{document}

when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z\rightarrow 0$$\end{document} in every fixed Stolz angle \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\arg (z)|\le \Delta $$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(0<\Delta <\pi /2)$$\end{document} .

Proof

This proof of the lemma can be straightforwardly done using Proposition 3.2 and Lemma 3.3; we omit the details. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Lemma 3.5

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x,y)\in S$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} G_{xy}\!\left( a,m; e^{-z/m}\zeta _{m}^h\right) = \left( {\textbf {1}}_{m\mid h}+O(|z|^{1/2})\right) G_{xy}\!\left( a,m;e^{-z/m}\right) , \end{aligned}$$\end{document}

when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z\rightarrow 0$$\end{document} in every fixed Stolz angle \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\arg (z)|\le \Delta $$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(0<\Delta <\pi /2)$$\end{document} .

Proof

This proof of the lemma can be straightforwardly done using Proposition 3.2, Lemmas 3.3 and 3.4. We sketch some of the details. For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q=e^{-z/m}\zeta _{m}^h$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\not \mid h$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&G_{01}\!\left( a,m;e^{-z/m}\zeta _{m}^h\right) \\&=\frac{(e^{-2z/m}\zeta _m^{2h};e^{-2z/m}\zeta _m^{2h})_\infty }{(e^{-z/m}\zeta _m^{h};e^{-z/m}\zeta _m^{h})_\infty } \frac{\sum _{n\ge 1}q^{m\frac{n(n-1)}{2}+an}}{(e^{-az/m}\zeta _{2m}^{m+2ah},e^{-(1-a/m)z}\zeta _{2m}^{-m-2ah}, e^{-z};e^{-z})_\infty }\\&\ll \left( |z|^{1/2}+{\textbf {1}}_{m\mid ah}\right) \exp \left( \frac{\gcd (m,h)^2}{6mz/\pi ^2}+\frac{\pi ^2}{6z}+ \frac{B_2\left( \left\{ \frac{2ah+m}{2m}\right\} \right) }{z/(2\pi ^2)}- \frac{\gcd (m,2h)^2}{12mz/\pi ^2}\right) \\&\ll \left( |z|^{1/2}+{\textbf {1}}_{m\mid ah}\right) G_{01}\!\left( a,m;e^{-z/m}\right) \exp \left( -\frac{2\pi ^2}{z} A_{01}(a,m; h)\right) , \end{aligned}$$\end{document}

where

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A_{01}(a,m; h)=\frac{m}{24}+\frac{\gcd (m,2h)^2}{24m}- \frac{\gcd (m,h)^2}{12m}-\frac{1}{12}- B_2\left( \left\{ \frac{2ah+m}{2m}\right\} \right) . \end{aligned}$$\end{document}

Similarly, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&G_{10}\!\left( a,m;e^{-z/m}\zeta _{m}^h\right) \\&=\frac{(\zeta _{m}^{ah}e^{-az/m},\zeta _{m}^{-ah}e^{-(1-a/m)z};e^{-z})_\infty }{(e^{-z};e^{-z})_\infty ^2( e^{-z/m}\zeta _m^{h};e^{-z/m}\zeta _m^h)_\infty } \sum _{k\ge 0}(-1)^{k}\frac{q^{m\frac{k(k+1)}{2}+mk+a}}{1-q^{mk+a}}\\&\ll \left( 1+|z|^{-1/2} {\textbf {1}}_{m\mid ah}\right) |z|\exp \left( \frac{\pi ^2}{3z}+\frac{\pi ^2\gcd (m,h)^2}{6mz}- \frac{2\pi ^2}{z}B_2\left( \left\{ \frac{a h}{m}\right\} \right) \right) \\&\ll \left( |z|^{1/2}+{\textbf {1}}_{m\mid ah}\right) G_{10}\!\left( a,m;e^{-z/m}\right) \exp \left( -\frac{2\pi ^2}{z} A_{10}(a,m; h)\right) , \end{aligned}$$\end{document}

where

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A_{10}(a,m; h)=\frac{m}{12}+B_2\left( \left\{ \frac{a h}{m}\right\} \right) - \frac{1}{6}-\frac{\gcd (m,h)^2}{12m}; \end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&G_{11}\!\left( a,m;e^{-z/m}\zeta _{m}^h\right) \\&=2\frac{(e^{-2z/m}\zeta _{m}^{2h};e^{-2z/m}\zeta _{m}^{2h})_\infty (e^{-2z};e^{-2z})_\infty ^2}{(e^{-z/m}\zeta _{m}^{h};e^{-z/m}\zeta _{m}^{h})_\infty ^2 (e^{-z};e^{-z})_\infty ^4}\\&\quad \times \frac{(e^{-az/m}\zeta _{m}^{ah},e^{-(1-a/m)z} \zeta _{m}^{-ah};e^{-z})_\infty }{(e^{-az/m}\zeta _{2m}^{m+2ah},e^{-(1-a/m)z}\zeta _{2m}^{-m-2ah}; e^{-z})_\infty }\sum _{n\ge 1}\frac{q^{an}}{1+q^{mn}}\\&\ll \left( |z|+{\textbf {1}}_{m\mid ah}\right) |z|^{1/2} \exp \left( \frac{\pi ^2\gcd (m,h)^2}{3mz}- \frac{\pi ^2\gcd (m,2h)^2}{12mz}+\frac{\pi ^2}{2z}\right) \\&\quad \times \exp \left( -\frac{2\pi ^2}{z} \left( B_2\left( \left\{ \frac{a h}{m}\right\} \right) - B_2\left( \left\{ \frac{2ah+m}{2m}\right\} \right) \right) \right) \\&\ll \left( |z|+{\textbf {1}}_{m\mid ah}\right) G_{11}\!\left( a,m;e^{-z/m}\right) \exp \left( -\frac{2\pi ^2}{z} A_{11}(a,m;h)\right) , \end{aligned}$$\end{document}

where

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A_{11}(a,m;h)=\frac{m}{8}+\frac{\gcd (m,2h)^2}{24m}+ B_2\left( \left\{ \frac{a h}{m}\right\} \right) - \frac{1}{4}-\frac{\gcd (m,h)^2}{6m}- B_2\left( \left\{ \frac{2ah+m}{2m}\right\} \right) . \end{aligned}$$\end{document}

It is clear that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{11}(a,m;h)=A_{01}(a,m;h)+A_{10}(a,m;h)$$\end{document} . Using elementary arguments, one can show that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{xy}(a,m;h)$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x,y)\in S$$\end{document} , is nonnegative for all integers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\not \mid h$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\ge 3$$\end{document} . Moreover, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\mid ah$$\end{document} but \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\not \mid h$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A_{01}(a,m; h)=\frac{m}{24}+\frac{\gcd (m,2h)^2}{24m}- \frac{\gcd (m,h)^2}{12m}\ge \frac{m}{24}-\frac{\gcd (m,h)^2}{24m}\ge \left( 1-\frac{1}{4}\right) \frac{m}{24}>0, \end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A_{10}(a,m; h)=\frac{m}{12}-\frac{\gcd (m,h)^2}{12m}\ge \left( 1-\frac{1}{4}\right) \frac{m}{12}>0, \end{aligned}$$\end{document}

because of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gcd (m,2h)\ge \gcd (m,h)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gcd (m,h)\le m/2$$\end{document} . The combination of all the above estimates proves the lemma. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Proof of Theorems 1.3 and 1.4

In this subsection, we will provide the proof of Theorem 1.3. To accomplish this, we will first establish Theorem 1.4, the residue class analogue of Ingham’s Tauberian theorem. A special case of Ingham’s Tauberian theorem, as presented in [14, p.1082, Eqs.(21),(22)], is stated as the following theorem.

Theorem 3.6

Let A(u) be an increasing function on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0, \infty )$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A(0)=0$$\end{document} be defined as the following Laplace transform:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f(s)=\int \limits _{0}^{\infty }e^{-us}\textrm{d}A(u). \end{aligned}$$\end{document}

If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(s)\sim f_0(s)$$\end{document} when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\rightarrow 0$$\end{document} in every fixed Stolz angle \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\arg (s)|\le \Delta $$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(0<\Delta <\pi /2)$$\end{document} , where

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f_0(s)=C(M/s)^{v\beta -\frac{1}{2}}e^{\beta ^{-1}(M/s)^{\beta }}\quad (\beta , M, C>0, v\in {\mathbb {R}}). \end{aligned}$$\end{document}

Then with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =\beta /(1+\beta )$$\end{document} , we have:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A(\omega )\sim \left( \frac{1-\alpha }{2\pi }\right) ^{\frac{1}{2}} C(M\omega )^{v\alpha -\frac{1}{2}}e^{\alpha ^{-1}(M\omega )^{\alpha }}, \text { as }\omega \rightarrow +\infty . \end{aligned}$$\end{document}

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(q)=\sum _{n\ge 0}c_nq^n$$\end{document} be a power series whose coefficients \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_n$$\end{document} are nonnegative and increasing, and converge absolutely for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|q|<1$$\end{document} . Notice that (3.11) involves a Riemann–Stieltjes integral. Therefore, if we define

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A(u)=c_{\lfloor u\rfloor }-c_{0}, \end{aligned}$$\end{document}

for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u\ge 0$$\end{document} , then A(u) is nonnegative and increasing with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A(0)=0$$\end{document} . Integration by parts for a Riemann–Stieltjes integral yields

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int \limits _{0}^{\infty }e^{-us}\textrm{d}A(u)=s\int \limits _{0}^{\infty }A(u)e^{-us}\textrm{d}u= \sum _{k\ge 0}A(k)\int \limits _{k}^{k+1}s e^{-us}\textrm{d}u&\nonumber \\ =(1-e^{-s})F(e^{-s})-c_0&. \end{aligned}$$\end{document}

As a conclusion, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(e^{-s})$$\end{document} meets the conditions in Theorem 3.6, then we can use that theorem to derive an asymptotic formula for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_n$$\end{document} .

However, in our case it is not easy to prove that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_n(a,m-a,m;x,y)$$\end{document} is increasing for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 0$$\end{document} . We only have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p_{n+m}(a,m-a,m;x,y)\ge p_{n}(a,m-a,m;x,y), \end{aligned}$$\end{document}

for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 0$$\end{document} , due to Lemma 2.1. Luckily, we can prove Theorem 1.4, a residue class analogue of Ingham’s Tauberian theorem, which works in our case.

Proof of Theorem 1.4

Due to condition (1) of Theorem 1.4, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_{mn+j}$$\end{document} is increasing in n, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\le j<m$$\end{document} . Moreover, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{n\ge 0}c_{mn+j}q^{n}=\frac{1}{m}\sum _{0\le h<m}e^{-2\pi \textrm{i}hj}q^{-j/m} f\!\left( q^{1/m}e^{2\pi \textrm{i}h/m}\right) , \end{aligned}$$\end{document}

by using the orthogonality of the m-th roots of unity. Using condition (2) of Theorem 1.4 and the above, with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{j}(u)=c_{m\lfloor u\rfloor +j}-c_j$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} c_j+\int \limits _{0}^{\infty }e^{-uz}\textrm{d}A_{j}(u)= (1-e^{-z})\sum _{n\ge 0}c_{mn+j}e^{-nz}\sim \frac{\alpha }{m^{1+\gamma }} z^{1+\gamma }e^{m^\rho \beta z^{-\rho }/\rho }, \end{aligned}$$\end{document}

by using (3.12), when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z\rightarrow 0$$\end{document} in every fixed Stolz angle \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\arg (z)|\le \Delta $$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(0<\Delta <\pi /2)$$\end{document} . Hence, Theorem 3.6 implies

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} c_{mn+j}\sim \frac{\alpha \beta ^{\frac{1+2\gamma }{2(1+\rho )}}}{\sqrt{2\pi (1+\rho )}}(mn)^{-\frac{1+2\gamma }{2(1+\rho )}- \frac{1}{2}}e^{(1+1/\rho )\beta ^{1/(1+\rho )} (mn)^{\rho /(1+\rho )}}, \end{aligned}$$\end{document}

as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\rightarrow +\infty $$\end{document} . Therefore, by using the simple estimates

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (n+r)^{\kappa _1}\sim n^{\kappa _1}\quad \text {and}\quad (n+r)^{\kappa _2}=n^{\kappa _2}+O(n^{\kappa _2-1})=n^{\kappa _2}+o(1), \end{aligned}$$\end{document}

as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\rightarrow \infty $$\end{document} , for any given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa _1\in {\mathbb {R}}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa _2<1$$\end{document} , we can replace mn by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$nm+j$$\end{document} in above asymptotic formula of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_f(mn+j)$$\end{document} . This completes the proof of Theorem 1.4. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

We now give the proof of Theorem 1.3. From inequality (3.13), Lemmas 3.4 and 3.5, we see that the generating function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{xy}(a,m;q)$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x,y)\in S$$\end{document} meets the conditions of Theorem 1.4. Thus, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p_{n}(a,m-a,m; x,y)\sim c_{a,m}(x){\widehat{p}}_n(x,y), \end{aligned}$$\end{document}

as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\rightarrow +\infty $$\end{document} , with

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left( c_{a,m}(0), c_{a,m}(1)\right) = \left( \frac{1}{2},~\frac{\psi \left( 1/2+a/2m\right) - \psi \left( {a}/{2m}\right) }{2\pi \csc (a\pi /m)}\right) . \end{aligned}$$\end{document}

Here,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\widehat{p}}_n(0,1)=\frac{e^{\pi \sqrt{n/3}}}{4\root 4 \of {3}n^{3/4}},\;\; {\widehat{p}}_n(1,0)= \frac{e^{2\pi \sqrt{n/6}}}{4\sqrt{3}n},\;\; \text {and}\;\; {\widehat{p}}_n(1,1)= \frac{1}{8n}e^{\pi \sqrt{n}}. \end{aligned}$$\end{document}

Recall the unrestricted partition function p(n) (see [13, Eq. (1.41)]), the distinct partition function q(n) (see [13, pages 109–110]), and the overpartition function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bar{p}}(n)$$\end{document} (see [14, Eq. (1)]), which have the following asymptotic formulas:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} q(n)\sim \frac{e^{\pi \sqrt{n/3}}}{4\root 4 \of {3}n^{3/4}}, \;\; p(n)\sim \frac{e^{2\pi \sqrt{n/6}}}{4\sqrt{3}n}, \;\; \text {and} \;\; {\bar{p}}(n)\sim \frac{1}{8n}e^{\pi \sqrt{n}}, \end{aligned}$$\end{document}

as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\rightarrow \infty $$\end{document} . This completes the proof of Theorem 1.3.

Remarks on residue class biases in distinct partitions

Theorem 1.1 provides results for biases in residue classes in partitions and overpartitions. For distinct partitions, we only have a partial result, given in Theorem 1.2. Beyond that partial result, we provide a conjecture, proposed in the end of this section.

In the following, we write \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_n(a,b;m):=p_n(a,b,m; 0,1)$$\end{document} for brevity. Then,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{n\ge 0}d_n(a,b;m)q^n= \sum _{\begin{array}{c} \lambda \in {{\mathcal {D}}}\\ \ell _{a,m}(\lambda )> \ell _{b,m}(\lambda ) \end{array}} q^{|\lambda |}. \end{aligned}$$\end{document}

If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b=m-a$$\end{document} , then Lemma 3.1 implies

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{n\ge 0}d_n(a,m-a;m)q^{n}=\frac{(-q;q)_\infty }{(-q^a,-q^{m-a},q^m;q^m)_\infty }\sum _{n\ge 1}q^{m\left( {\begin{array}{c}n\\ 2\end{array}}\right) +n a}. \end{aligned}$$\end{document}

From Equation (4.1), we have the following inequalities for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_{n}(1,2;3)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_{n}(2,1;3)$$\end{document} .

Proposition 4.1

We have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} d_{3n+2}(1,2;3)\le d_{3n+2}(2,1;3), \\ d_{3n}(1,2;3)\ge d_{3n}(2,1;3), \quad \text {and}\quad d_{3n+1}(1,2;3)\ge d_{3n+1}(2,1;3), \end{aligned}$$\end{document}

for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 0$$\end{document} .

Proof

The proof is straightforward and uses a dissection strategy. First, we observe

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{n\ge 0}(d_{n}(1,2;3)-d_{n}(2,1;3))q^n= \frac{(-q^3;q^3)_\infty }{(q^3;q^3)_\infty } \sum _{n\ge 1}q^{\frac{n(3n-1)}{2}}(1-q^{n}). \end{aligned}$$\end{document}

Dissection of the last sum gives

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\sum _{n\ge 0}q^{\frac{n(3n-1)}{2}}(1-q^{n}) =\sum _{0\le j\le 2}\sum _{n\ge 0}q^{\frac{3(3n+j)-3n+j)}{2}}(1-q^{3n+j})\\&=\sum _{n\ge 0}q^{\frac{3n(9n-1)}{2}}(1-q^{3n})+ q\sum _{n\ge 0}q^{\frac{9n(3n+5)}{2}}(1-q^{12n+6})- q^2\sum _{n\ge 0}q^{\frac{9n(3n+7)}{2}}(1-q^{6n+3}). \end{aligned}$$\end{document}

This completes the proof. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

A similar argument can be applied to compare \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_{n}(1,3;4)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_{n}(3,1;4)$$\end{document} :

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{n\ge 0}\left( d_n(1,3;4)-d_n(3,1;4)\right) q^{n} =\frac{(-q;q)_\infty }{(-q,-q^{3},q^4;q^4)_\infty } \sum _{n\ge 1}q^{2n^2-n}(1-q^{n})&\\ =\frac{1}{(q^2;q^2)_\infty }\sum _{n\ge 1}q^{2n^2-n}(1-q^{2n}) =\frac{1}{(q^4;q^2)_\infty }\sum _{n\ge 1}q^{2n^2}\sum _{0\le j<n}q^{2j-n}&. \end{aligned}$$\end{document}

Therefore, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_n(1,3;4)\ge d_n(3,1;4)$$\end{document} . This is also a special case of Theorem 1.2.

We turn to the general case. For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le a<b\le m$$\end{document} , by Equation (2.1) we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{n\ge 0}d_{n}(a,b;m)q^n= \frac{(-q;q)_\infty }{(-q^a,-q^b;q^m)_\infty } \sum _{\begin{array}{c} j> k\\ j,k\ge 0 \end{array}}\frac{q^{\frac{1}{2}m(j(j-1)+k(k-1))+aj+bk}}{(q^m;q^m)_{j}(q^m;q^m)_{k}}. \end{aligned}$$\end{document}

Based on this generating function and extensive computer experiments, we propose the following conjecture.

Conjecture 4.1

For any positive integers a, b, m such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le a<b\le m$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\ge 3$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(a,b,m)\ne (1,2,3)$$\end{document} , there exists a constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{a,b,m}>0$$\end{document} such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} d_{n}(a,b; m)\ge d_{n}(b,a; m),\quad \text {for all}\quad n\ge N_{a,b, m}. \end{aligned}$$\end{document}

Moreover,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} N_{a,m-a,m}=0,\quad \text {for all}\quad 1\le a<m/2, \end{aligned}$$\end{document}

except

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} N_{2,3,5}=45, N_{2,4,6}=5, N_{3,4,7}=8, N_{4,5,9}=9. \end{aligned}$$\end{document}

Remark 4.1

The constants \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{a,b, m}$$\end{document} appear to be all relatively small. In particular, there exists an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_0>9$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{a,b, m}$$\end{document} equals zero for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\ge m_0$$\end{document} .

Remark 4.2

We note that we confirmed Conjecture 4.1 for a class of special cases in Theorem 1.2.