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2- and 3-Dissections of Second-, Sixth-, and Eighth-Order Mock Theta Functions

Frank Garvan, Hemjyoti Nath

Abstract

In this paper, we develop a unified method for obtaining and proving $m$-dissections of mock theta functions. Our approach builds upon a transformation formula for Appell--Lerch sums due to Hickerson and Mortenson, which allows these sums to be expressed as linear combinations of Appell--Lerch sums together with suitable theta products. By systematically exploiting this representation, and through extensive symbolic computations carried out in Maple, we derive explicit dissection identities in a direct and effective manner. We focus exclusively on the cases of $2$- and $3$-dissections.

2- and 3-Dissections of Second-, Sixth-, and Eighth-Order Mock Theta Functions

Abstract

In this paper, we develop a unified method for obtaining and proving -dissections of mock theta functions. Our approach builds upon a transformation formula for Appell--Lerch sums due to Hickerson and Mortenson, which allows these sums to be expressed as linear combinations of Appell--Lerch sums together with suitable theta products. By systematically exploiting this representation, and through extensive symbolic computations carried out in Maple, we derive explicit dissection identities in a direct and effective manner. We focus exclusively on the cases of - and -dissections.

Paper Structure

This paper contains 11 sections, 37 theorems, 76 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Second-Order Mock Theta Functions
  4. Sixth-Order Mock Theta Functions
  5. Eighth-Order Mock Theta Functions
  6. Preliminary Theta Function Identities
  7. A General Algorithm for Proving Theta Identities
  8. 2-Dissections of Mock Theta Functions
  9. 3-Dissections of Mock Theta Functions
  10. General Algorithm for $m$-Dissections
  11. Concluding remarks

Key Result

Lemma 2.1

The following theta function identities hold:

Theorems & Definitions (49)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Definition 2.4: hickerson2014hecke
  • Proposition 2.5: hickerson2014hecke
  • Theorem 2.6: hickerson2014hecke
  • Corollary 2.7: hickerson2014hecke
  • Lemma 3.1
  • ...and 39 more