Ramanujan's $_{1}\psi_1$ summation, Hecke-type double sums, and Appell-Lerch sums

TL;DR

This paper presents a new proof connecting Ramanujan's $_{1}\psi_1$ summation with Hecke-type double sums, Appell-Lerch sums, and theta functions, enhancing understanding of these special functions.

Contribution

It offers a novel proof of a recent formula relating Hecke-type double sums to Appell-Lerch sums using Ramanujan's $_{1}\psi_1$ summation.

Findings

New proof of Hickerson and Mortenson's formula

Expanded understanding of relations between special sums and theta functions

Enhanced techniques for manipulating Ramanujan's summation

Abstract

We use a specialization of Ramanujan's ${}_1\psi_1$ summation to give a new proof of a recent formula of Hickerson and Mortenson which expands a special family of Hecke-type double sums in terms of Appell-Lerch sums and theta functions.