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

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.