A simple proof of a congruence for a series involving the little $q$-Jacobi polynomials

TL;DR

This paper provides a straightforward and explicit proof of a modulo 4 congruence related to a series involving little q-Jacobi polynomials, originating from overpartition function research.

Contribution

It introduces a simple, explicit proof of a specific congruence for a series with little q-Jacobi polynomials, enhancing understanding of related partition functions.

Findings

Proves a mod 4 congruence for a series involving little q-Jacobi polynomials

Simplifies previous proofs with a more direct approach

Connects polynomial series to overpartition functions

Abstract

We give a simple and a more explicit proof of a mod $4$ congruence for a series involving the little $q$-Jacobi polynomials which arose in a recent study of a certain restricted overpartition function.

Equations

$$p_{n}(x;\alpha,\beta:q):={}_{2}\phi_{1}\bigg{(}\begin{matrix}q^{-n},&\alpha\beta q^{n+1}\\ &\alpha q\end{matrix}\,;qx\bigg{)},$$

$${}_{r+1}{\varphi }_r\left(\begin{array}{c}{a}_1,a_2,\dots ,a_{r+1}\\ b_1,b_2,\dots ,b_r\end{array};q,z\right):=\sum _{n=0}^{\infty }\frac{(a_1;q{)}_n(a_2;q{)}_n\cdots (a_{r+1};q{)}_n}{(q;q{)}_n(b_1;q{)}_n\cdots (b_r;q{)}_n}{z}^n,$$

$$(A;q{)}_0=1;(A;q{)}_n=(1-A)(1-Aq)\cdots (1-A{q}^{n-1}),n\ge 1,$$

$$(A;q{)}_{\infty }=\underset{n\to \infty }{\lim }(A;q{)}_n(|q|<1).$$

$${\overline{P}}_{\omega }(q):=\sum _{n=1}^{\infty }{\overline{p}}_{\omega }(n)q^n$$

$$+\frac{(-q;q{)}_{\infty }}{(q;q{)}_{\infty }}\sum _{n=0}^{\infty }\frac{(q;q^2{)}_n(-q{)}^n}{(-q;q^2{)}_n(1+q^{2n})}{p}_{2n}(-1;q^{-2n-1},-1:q).$$

$$\sum _{n=0}^{\infty }\frac{(q;q^2{)}_n(-q{)}^n}{(-q;q^2{)}_n(1+q^{2n})}{p}_{2n}(-1;q^{-2n-1},-1:q)\equiv \frac{1}{2}\frac{(q;q^2{)}_{\infty }}{(-q;q^2{)}_{\infty }}(\mathrm{mod}4).$$

$$\sum _{n=0}^{\infty }\frac{(q;q^2{)}_n(-q{)}^n}{(-q;q^2{)}_n(1+q^{2n})}{p}_{2n}(-1;q^{-2n-1},-1:q)$$

$$=\frac{1}{2}\frac{(q;q^2{)}_{\infty }}{(-q;q^2{)}_{\infty }}+\frac{4q^2}{(1+q)}\sum _{n=0}^{\infty }\frac{(q^3;q^2{)}_n(-q{)}^n}{(-q^3;q^2{)}_n(1+q^{2n+2})}\sum _{j=0}^n\frac{(-q;q{)}_{2j}{q}^{2j}}{(q^2;q{)}_{2j}}.$$

$$\sum _{n=0}^{\infty }\frac{(q;q^2{)}_n(-q{)}^n}{(-q;q^2{)}_n(1+q^{2n})}{p}_{2n}(-1;q^{-2n-1},-1:q)=\sum _{n=0}^{\infty }\frac{(q;q^2{)}_n(-q{)}^n}{(-q;q^2{)}_n(1+q^{2n})}\sum _{j=0}^{2n}\frac{(-1;q{)}_j}{(q;q{)}_j}(-q{)}^j.$$

$$A(q):=\sum _{n=0}^{\infty }\frac{(q;q^2{)}_n(-q{)}^n}{(-q;q^2{)}_n(1+q^{2n})}\sum _{j=0}^{2n}\frac{(-1;q{)}_j}{(q;q{)}_j}{q}^j.$$

$$\frac{(abq;q{)}_n}{(bq;q{)}_n}=1+b(1-a)\sum _{j=1}^n\frac{(abq;q{)}_{j-1}{q}^j}{(bq;q{)}_j}.$$

$$\sum _{j=0}^{2n}\frac{(-1;q{)}_j{q}^j}{(q;q{)}_j}=\frac{(-q;q{)}_{2n}}{(q;q{)}_{2n}}.$$

$$A(q)$$

$$=\frac{1}{2}+\sum _{n=1}^{\infty }\frac{(-q^2;q^2{)}_{n-1}}{(q^2;q^2{)}_n}(-q{)}^n$$

$$=\frac{1}{2}\frac{(q;q^2{)}_{\infty }}{(-q;q^2{)}_{\infty }},$$

$$\sum _{n=0}^{\infty }\frac{(q;q^2{)}_n(-q{)}^n}{(-q;q^2{)}_n(1+q^{2n})}{p}_{2n}(-1;q^{-2n-1},-1:q)-A(q)$$

$$=\sum _{n=0}^{\infty }\frac{(q;q^2{)}_n(-q{)}^n}{(-q;q^2{)}_n(1+q^{2n})}\sum _{j=0}^{2n}((-1{)}^j-1)\frac{(-1;q{)}_j{q}^j}{(q;q{)}_j}$$

$$=-2\sum _{n=0}^{\infty }\frac{(q;q^2{)}_n(-q{)}^n}{(-q;q^2{)}_n(1+q^{2n})}\sum _{j=1}^n\frac{(-1;q{)}_{2j-1}{q}^{2j-1}}{(q;q{)}_{2j-1}}$$

$$=\frac{4q^2}{(1+q)}\sum _{n=0}^{\infty }\frac{(q^3;q^2{)}_n(-q{)}^n}{(-q^3;q^2{)}_n(1+q^{2n+2})}\sum _{j=0}^n\frac{(-q;q{)}_{2j}{q}^{2j}}{(q^2;q{)}_{2j}}.$$

Full text

1. Introduction

In [3], Andrews, Schultz, Yee and the author studied the overpartition function $\overline{p}_{\omega}(n)$, namely, the number of overpartitions of $n$ such that all odd parts are less than twice the smallest part, and in which the smallest part is always overlined. In the same paper, they obtained a representation for the generating function of $\overline{p}_{\omega}(n)$ in terms of a ${}_{3}\phi_{2}$ basic hypergeometric series and an infinite series involving the little $q$-Jacobi polynomials. The latter are given by [2, Equation (3.1)]

[TABLE]

where the basic hypergeometric series ${}_{r+1}\phi_{r}$ is defined by

[TABLE]

and where we use the notation

[TABLE]

The precise representation for the generating function of $\overline{p}_{\omega}(n)$ obtained in [3] is as follows.

Theorem 1.1**.**

The following identity holds for $|q|<1$:

[TABLE]

Later, Bringmann, Jennings-Shaffer and Mahlburg [4, Theorem 1.1] showed that $\overline{P}_{\omega}(q)+\frac{1}{4}-\frac{\eta(4\tau)}{2\eta(2\tau)^{2}}$, where $q=e^{2\pi i\tau}$ and $\eta(\tau)$ is the Dedekind eta function, can be completed to a function $\hat{P}_{\omega}(\tau)$, which transforms like a weight $1$ modular form. They called the function $\overline{P}_{\omega}(q)+\frac{1}{4}-\frac{\eta(4\tau)}{2\eta(2\tau)^{2}}$ a higher depth mock modular form.

While the series involving the little $q$-Jacobi polynomials in Theorem 1.1 itself looks formidable, it was shown in [3, Theorem 1.3] that modulo $4$, it is a simple $q$-product. The mod $4$ congruence proved in there is given below.

Theorem 1.2**.**

The following congruence holds:

[TABLE]

The proof of this congruence in [3] is beautiful but somewhat involved. The objective of this short note is to give a very simple proof of it. In fact, we derive it as a trivial corollary of the following result.

Theorem 1.3**.**

For $|q|<1$, we have

[TABLE]

The presence of $4$ in front of the series on the right-hand side in the above equation immediately implies that Theorem 1.2 holds.

2. Proof of Theorem 1.3

Observe that from (1.1),

[TABLE]

However, let us first consider

[TABLE]

The only difference in the series on the right-hand side of (2.1) and the series in (2.2) is the presence of $(-1)^{j}$ inside the finite sum in the former.

To simplify $A(q)$, we start with a result of Alladi [1, p. 215, Equation (2.6)]:

[TABLE]

Let $a=-1,b=1$ and replace $n$ by $2n$ so that

[TABLE]

Substitute (2.4) in (2.2) to see that

[TABLE]

where in the last step we used the $q$-binomial theorem $\sum_{n=0}^{\infty}\frac{(a;q)_{n}}{(q;q)_{n}}z^{n}=\frac{(az;q)_{\infty}}{(z;q)_{\infty}}$, valid for $|z|<1$ and $|q|<1$.

From (2.1) and (2.2),

[TABLE]

Invoking (2), we see that the proof of Theorem 1.3 is complete.