A study on Horn matrix functions and its confluent cases

TL;DR

This paper extends Horn's hypergeometric functions to matrix versions, exploring their properties, differential equations, convergence, and integral representations, thereby completing the matrix generalization of Horn's list of hypergeometric series.

Contribution

It introduces the matrix version of Horn's hypergeometric functions and their confluent cases, along with their properties and integral representations, expanding the theory to matrix functions.

Findings

Defined matrix Horn functions and confluent cases

Derived differential equations and convergence regions

Provided integral representations of these matrix functions

Abstract

In this paper, we give the matrix version of Horn's hypergeometric function and its confluent cases. We also discuss the regions of convergence, the system of matrix differential equations of bilateral type, differential formulae and infinite summation formulae satisfied by these hypergeometric matrix functions. We also give the certain integral representation of these hypergeometric matrix functions. The study of these 23 matrix functions leads to completing the matrix generalization of Horn's list of 34 hypergeometric series.