Extended Beta and Gamma Matrix Functions via 2-Parameter Mittag-Leffler Matrix Function

Rahul Goyal,Georgia Irina Oros,Shilpi Jain,Praveen Agarwal

doi:10.3390/math10060892

Rahul Goyal, Georgia Irina Oros + Show 2 more

Open Access

https://doi.org/10.3390/math10060892

Copy DOI

Abstract

The main aim of this article is to study an extension of the Beta and Gamma matrix functions by using a two-parameter Mittag-Leffler matrix function. In particular, we investigate certain properties of these extended matrix functions such as symmetric relation, integral representations, summation relations, generating relation and functional relation.

Highlights

Introduction and PreliminariesThe theory of special matrix functions has been introduced by Jó and Cortés [1,2] and they proved some basic and important properties of the Gamma and Beta matrix functions and a limit expression for the Gamma function of a matrix
After some time using these results, they studied the hypergeometric function with matrix arguments
Motivated by the investigations of the extended Gamma, Beta, and Gauss hypergeometric matrix functions given in [1,2] many researchers [3–7] were attracted to work in the field of special functions with matrix arguments

Summary

Introduction

The theory of special matrix functions has been introduced by Jó and Cortés [1,2] and they proved some basic and important properties of the Gamma and Beta matrix functions and a limit expression for the Gamma function of a matrix. After some time using these results, they studied the hypergeometric function with matrix arguments. Motivated by the investigations of the extended Gamma, Beta, and Gauss hypergeometric matrix functions given in [1,2] many researchers [3–7] were attracted to work in the field of special functions with matrix arguments. Special matrix functions are found in the solutions for some physical problems and applications of these functions increased in the statistics [8], Lie group theory [9] and differential equations. Theoretical physics and the emerging theory of orthogonal matrix polynomials [10–12], these special matrix functions play a vital role due to their applications. To discuss our main results we require prior knowledge of some special matrix functions

Objectives

Results

Conclusion