On the matrix $q$-Kummer equation and its solutions

TL;DR

This paper develops a general theory for second-order matrix bilateral difference equations, introduces the matrix q-Kummer equation and its series and integral solutions, and discusses related matrix Gauss equations.

Contribution

It introduces the matrix q-Kummer equation of bilateral type and provides explicit series and integral solutions, advancing the understanding of matrix difference equations.

Findings

Derived the series solution for the matrix q-Kummer equation.

Obtained integral solutions and solutions at infinity.

Explored the matrix Gauss difference equation and its solutions.

Abstract

In the present paper, a general theory for the second-order matrix difference equation of bilateral type is discussed. We introduced the matrix $q$-Kummer equation of bilateral type and presented the $q$-Kummer matrix function as a series solution. Later, we obtain the integral solutions of this Kummer equation and solution at $\infty$. We also give a brief idea about the matrix Gauss difference equation of bilateral type and its solutions.