On integral representations of $q$-difference operators and their applications

TL;DR

This paper presents integral representations of two $q$-difference operators using special functions related to asymptotic solutions of $q$-difference equations, unified via the $(p,q)$-differential operator and its kernel function.

Contribution

It introduces unified integral representations of $q$-difference operators through the $(p,q)$-differential operator and provides a kernel function for generating $(p,q)$-factorials.

Findings

Integral representations expressed in terms of special functions.

Unified framework via the $(p,q)$-differential operator.

Kernel function generating $(p,q)$-factorials.

Abstract

Integral representations of two $q$-difference operators are provided in terms of special functions arising in the theory of asymptotic solutions to $q$-difference equations in the complex domain. Both representations are unified through the so-called $(p,q)$-differential operator, for which a kernel-like function is provided, generating the sequence of $(p,q)$-factorials.