Proof of a conjecture of Andrews and Bachraoui on a Hecke sum

TL;DR

This paper proves a conjecture connecting a generating function for two-color partitions to a Hecke-type sum, using indefinite theta functions and mock theta function transformations.

Contribution

It provides a proof of a conjecture linking partition generating functions to Hecke sums via advanced modular and theta function techniques.

Findings

Confirmed the conjecture of Andrews and Bachraoui.

Established a new connection between partition functions and Hecke sums.

Applied Zwegers' theory to prove the conjecture.

Abstract

In this paper, we prove a conjecture of Andrews and Bachraoui relating a generating function arising from two-color partitions (with odd smallest part and restrictions on the even parts) to a Hecke-type double sum. Our proof is based on Zwegers' theory of indefinite theta functions together with modular transformation properties of mock theta functions.