A certain generalization of $q$-hypergeometric functions and their related monodromy preserving deformation II

We define a nonlinear $q$-difference system $mathcal{P}_{N,(M_{-},M_{+})}$ as monodromy preserving deformations of a certain linear equation. We study its relation to a series $mathcal{F}_{N,M}$ defined as a certain generalization of $q$-hypergeometric functions.