On the dual nature of partial theta functions and Appell-Lerch sums

TL;DR

This paper explores the duality between partial theta functions and Appell-Lerch sums, establishing a framework that relates identities involving these functions and constructing bilateral q-series with mixed mock modular properties.

Contribution

It introduces a duality theory connecting partial theta functions and Appell-Lerch sums, and applies it to construct new bilateral q-series with mixed mock modular behavior.

Findings

Established a duality between partial theta functions and Appell-Lerch sums.

Related identities involving these functions through the duality framework.

Constructed bilateral q-series exhibiting mixed mock modular properties.

Abstract

In recent work, Hickerson and the author demonstrated that it is useful to think of Appell--Lerch sums as partial theta functions. This notion can be used to relate identities involving partial theta functions with identities involving Appell--Lerch sums. In this sense, Appell--Lerch sums and partial theta functions appear to be dual to each other. This duality theory is not unlike that found by Andrews between various sets of identities of Rogers-Ramanujan type with respect to Baxter's solution to the hard hexagon model of statistical mechanics. As an application we construct bilateral $q$-series with mixed mock modular behaviour.