On $q$-Middle Convolution and $q$-Hypergeometric Equations

TL;DR

This paper reformulates $q$-integral transformations linked to $q$-middle convolution, analyzes their convergence, and derives $q$-integral solutions for variants of the $q$-hypergeometric equation.

Contribution

It introduces a reformulation of $q$-integral transformations related to $q$-middle convolution and applies this to obtain new $q$-integral solutions for $q$-hypergeometric equations.

Findings

Convergence conditions for $q$-integral transformations established.

New $q$-integral representations of solutions derived.

Enhanced understanding of $q$-middle convolution's role in $q$-hypergeometric equations.

Abstract

The $q$-middle convolution was introduced by Sakai and Yamaguchi. In this paper, we reformulate $q$-integral transformations associated with the $q$-middle convolution. In particular, we discuss convergence of the $q$-integral transformations. As an application, we obtain $q$-integral representations of solutions to the variants of the $q$-hypergeometric equation by applying the $q$-middle convolution.