Partial theta functions and mock modular forms as q-hypergeometric series

TL;DR

This paper explores the analytic properties of partial theta functions and mock theta functions as q-hypergeometric series, revealing their relationships and distinctions across different convergence domains.

Contribution

It provides new insights into the connection between mock theta functions and partial theta functions within the framework of q-hypergeometric series, expanding understanding of their analytic behavior.

Findings

Mock theta functions can be represented as q-hypergeometric series in one domain.

Partial theta functions relate to these series in a different convergence domain.

The series exhibit equality in one domain and are related by partial theta functions in another.

Abstract

Ramanujan studied the analytic properties of many $q$-hypergeometric series. Of those, mock theta functions have been particularly intriguing, and by work of Zwegers, we now know how these curious $q$-series fit into the theory of automorphic forms. The analytic theory of partial theta functions however, which have $q$-expansions resembling modular theta functions, is not well understood. Here we consider families of $q$-hypergeometric series which converge in two disjoint domains. In one domain, we show that these series are often equal to one another, and define mock theta functions, including the classical mock theta functions of Ramanujan, as well as certain combinatorial generating functions, as special cases. In the other domain, we prove that these series are typically not equal to one another, but instead are related by partial theta functions.