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

Joshua Males, Andreas Mono, Larry Rolen

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.

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

Abstract

In the theory of harmonic Maass forms and mock modular forms, mock theta functions are distinguished examples which arose from -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 -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 -hypergeometric series.

Paper Structure

This paper contains 6 sections, 7 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $q$-hypergeometric series
  4. Higher depth mock modular forms
  5. Modular completions
  6. Proof of Theorems \ref{['Theorem: main']} and \ref{['Theorem main: completions']}

Key Result

Theorem 1.1

Let $\zeta$ be a root of unity. Then the functions $f_j$ for $j=1,2,3$ are each mock theta functions of depth two. Furthermore, we have the following representations as double-sum $q$-series.

Theorems & Definitions (14)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1: sriva
  • Lemma 2.2: fine
  • Definition
  • Definition : raum
  • Definition
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 4 more