A solution in terms of mock modular forms for the $q$-Painlev\'{e} equation of the type $(A_2+A_1)^{(1)}$

TL;DR

This paper expresses solutions to a specific $q$-Painlevé equation using mock modular forms, specifically the $unction$, highlighting a deep connection between mock modular forms and integrable systems.

Contribution

It introduces a novel solution to the $(A_2+A_1)^{(1)}$ $q$-Painleve9 equation via the $unction$, linking mock modular forms with discrete integrable systems.

Findings

Solution expressed in terms of the $unction$

Suggests a close relationship between mock modular forms and $ au$-functions

Provides new insights into the structure of $q$-Painleve9 equations

Abstract

We present a solution of the $(A_2+A_1)^{(1)}$ $q$-Painlev\'{e} equation in terms of the $\mu$-function. The $\mu$-function introduced by Zwegers is the most fundamental object in the study of mock theta functions. The results of this paper give us an expectation that the theory of mock modular forms and the $\tau$-functions of discrete integrable systems are closely related.