Hecke-type double sums, Appell-Lerch sums, and mock theta functions (I)

TL;DR

This paper develops a new formula connecting Hecke-type double sums, Appell-Lerch sums, and theta functions, providing proofs for classical identities and mock theta conjectures, with applications to affine Lie algebra representations.

Contribution

It introduces a novel formula linking partial theta functions with Appell-Lerch sums, enabling new proofs of classical and mock theta function identities.

Findings

Proved classical Hecke-type double sum identities.

Provided a new proof of the mock theta conjectures.

Applied the formula to affine Lie algebra string functions.

Abstract

By developing a connection between partial theta functions and Appell-Lerch sums, we find and prove a formula which expresses Hecke-type double sums in terms of Appell-Lerch sums and theta functions. Not only does our formula prove classical Hecke-type double sum identities such as those found in work Kac and Peterson on affine Lie Algebras and Hecke modular forms, but once we have the Hecke-type forms for Ramanujan's mock theta functions our formula gives straightforward proofs of many of the classical mock theta function identities. In particular, we obtain a new proof of the mock theta conjectures. Our formula also applies to positive-level string functions associated with admissable representations of the affine Lie Algebra $A_1^{(1)}$ as introduced by Kac and Wakimoto.