Jacobi's triple product, mock theta functions, unimodal sequences and the $q$-bracket

TL;DR

This paper reveals the natural emergence of the universal mock theta function from the Jacobi triple product's reciprocal, linking it to unimodal sequences, modular forms, and quantum behaviors, with new formulas at roots of unity.

Contribution

It demonstrates that the universal mock theta function $g_3(z,q)$ arises from the Jacobi triple product and connects it to rank functions and the $q$-bracket, providing new formulas and insights.

Findings

$g_3(z,q)$ extends to the complex plane minus the unit circle.

Finite formulas for $g_3(z,q)$ at roots of unity are derived.

Mock theta functions exhibit quantum behaviors inside, outside, and on the unit circle.

Abstract

In Ramanujan's final letter to Hardy, he listed examples of a strange new class of infinite series he called "mock theta functions". It turns out all of these examples are essentially specializations of a so-called universal mock theta function $g_3(z,q)$ of Gordon-McIntosh. Here we show that $g_3$ arises naturally from the reciprocal of the classical Jacobi triple product -- and is intimately tied to rank generating functions for unimodal sequences, which are connected to mock modular and quantum modular forms -- under the action of an operator related to statistical physics and partition theory, the $q$-bracket of Bloch-Okounkov. Secondly, we find $g_3(z,q)$ to extend in $q$ to the entire complex plane minus the unit circle, and give a finite formula for this universal mock theta function at roots of unity, that is simple by comparison to other such formulas in the literature; we also indicate similar formulas for other $q$-hypergeometric series. Finally, we look at interesting "quantum" behaviors of mock theta functions inside, outside, and on the unit circle.