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

We present a solution of the $(A_2+A_1{)}^{(1)}$ $q$-Painlevé 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.