Modular Transformations of Ramanujan's Tenth Order Mock Theta Functions

TL;DR

This paper computes the modular transformations of Ramanujan's tenth order mock theta functions using Hecke-type identities and Zwegers' results, confirming previous numerical work and recent conjectures.

Contribution

It provides explicit modular transformation formulas for these mock theta functions, advancing understanding of their structure and confirming prior numerical and conjectural findings.

Findings

Confirmed numerical transformations by Gordon and McIntosh

Validated recent conjecture by Cheng, Duncan, and Harvey

Derived explicit completions and shadows for the functions

Abstract

The modular transformations of Ramanujan's tenth order mock theta functions are computed, beginning from Choi's Hecke-type identites and using Zwegers' results on indefinite theta series. Explicit completions and shadows are found as an intermediate step. Our result for the modular transformations confirms numerical work by Gordon and McIntosh, and a recent conjecture by Cheng, Duncan, and Harvey.