Unimodal sequences and mixed false theta functions

TL;DR

This paper explores generalized generating functions for unimodal sequences, expressing them through mixed false theta functions, and connects these results to classical identities and recent work on mock modular forms.

Contribution

It introduces two-parameter generalizations of unimodal sequence generating functions involving mixed false theta functions, extending recent research on mock modularity.

Findings

Explicit representations involving mixed false theta functions

Recovery of classical partial theta identities

Connections to recent mock modularity research

Abstract

We consider two-parameter generalizations of Hecke-Appell type expansions for the generating functions of unimodal and special unimodal sequences. We then determine their explicit representations which involve mixed false theta functions. These results complement recent striking work of Mortenson and Zwegers on the mixed mock modularity of the generalized $U$-function due to Hikami and Lovejoy. As an application, we demonstrate how to recover classical partial theta function identities which appear in Ramanujan's lost notebook and in work of Warnaar.