Higher depth mock theta functions and $q$-hypergeometric series

TL;DR

This paper introduces higher depth mock theta functions, extending classical mock theta functions, and develops their $q$-hypergeometric representations, modular completions, and specific examples of depth two functions.

Contribution

It defines higher depth mock theta functions, provides explicit $q$-hypergeometric formulas, and connects them to modular completions, advancing the understanding of mock modular forms.

Findings

Presented three explicit depth two mock theta functions.

Derived $q$-hypergeometric series representations for these functions.

Established their modular completions and relations to classical forms.

Abstract

In the theory of harmonic Maass forms and mock modular forms, mock theta functions are distinguished examples which arose from $q$-hypergeometric examples of Ramanujan. Recently, there has been a body of work on higher depth mock modular forms. Here, we introduce distinguished examples of these forms which we call higher depth mock theta functions and develop $q$-hypergeometric expressions for them. We provide three examples of mock theta functions of depth two, each arising by multiplying a classical mock theta function with a certain specialization of a universal mock theta function. In addition, we give their modular completions, and relate each to a $q$-hypergeometric series.