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

TL;DR

This paper introduces a new generalization of $q$-hypergeometric functions, explores their duality, and analyzes associated $q$-difference nonlinear equations with specific solutions.

Contribution

It defines the series $ ext{F}_{M,N}$ as a novel generalization of $q$-hypergeometric functions and investigates their duality and related nonlinear $q$-difference equations.

Findings

Introduction of the $ ext{F}_{M,N}$ series as a new $q$-hypergeometric generalization.

Identification of duality properties of $ ext{F}_{M,N}$.

Derivation of $q$-difference nonlinear equations with particular solutions.

Abstract

We define a series $\mathcal{F}_{M,N}$ as a certain generalization of $q$-hypergeometric function. We study its duality and the system of $q$-difference nonlinear equations which admits particular solutions in terms of $\mathcal{F}_{1,M}$.