Proofs of Some Conjectures of Chan on Appell-Lerch Sums

TL;DR

This paper proves several conjectured and new congruences for coefficients related to Ramanujan's Appell-Lerch sums, connecting them to mock theta functions and providing explicit generating function representations.

Contribution

It establishes new congruences for coefficients of Appell-Lerch sums and provides explicit generating function formulas, confirming conjectures and extending previous results.

Findings

Proved conjectured congruences modulo 25 for certain coefficients.

Established new congruences modulo 125 for coefficients.

Derived explicit generating functions for specific coefficient subsequences.

Abstract

On page 3 of his lost notebook, Ramanujan defines the Appell-Lerch sum $$\phi(q):=\sum_{n=0}^\infty \dfrac{(-q;q)_{2n}q^{n+1}}{(q;q^2)_{n+1}^2},$$ which is connected to some of his sixth order mock theta functions. Let $\sum_{n=1}^\infty a(n)q^n:=\phi(q)$. In this paper, we find a representation of the generating function of $a(10n+9)$ in terms of $q$-products. As corollaries, we deduce the congruences $a(50n+19)\equiv a(50n+39)\equiv a(50n+49)\equiv0~(\textup{mod}~25)$ as well as $a(1250n+250r+219)\equiv 0~(\textup{mod}~125)$, where $r=1$, $3$, and $4$. The first three congruences were conjectured by Chan in 2012, whereas the congruences modulo 125 are new. We also prove two more conjectural congruences of Chan for the coefficients of two Appell-Lerch sums.