On the Extended Hypergeometric Matrix Functions and Their Applications for the Derivatives of the Extended Jacobi Matrix Polynomial

Fuli He,M Abdalla,Ahmed Bakhet,M Hidan

doi:10.1155/2020/4268361

Abstract

In this paper, we obtain some generating matrix functions and integral representations for the extended Gauss hypergeometric matrix function EGHMF and their special cases are also given. Furthermore, a specific application for the extended Gauss hypergeometric matrix function which includes Jacobi matrix polynomials is constructed.

Highlights

Generalizations of the classical special functions to matrix setting have become important during the previous years
New extensions of some of the wellknown special matrix functions such as gamma matrix function, beta matrix function, and Gauss hypergeometric matrix function have been extensively studied in recent papers [5,6,7,8,9,10]
E main object of this paper is to investigate various properties for the extended Gauss hypergeometric matrix function EGHMF. e generating functions and integral formulas are derived for EGHMF

Summary

Introduction

Generalizations of the classical special functions to matrix setting have become important during the previous years. A specific application for the extended Gauss hypergeometric matrix function which includes Jacobi matrix polynomials is constructed. Let A and B be positive stable matrices in Cr×r; the generalized Gamma matrix function Γ(A, B) is defined by. Suppose that A, B, and P are positive stable and commutative matrices in Cr×r satisfying spectral condition (1); the extended beta matrix function. Abdalla and Bakhet [8] used B(A, B; P) to extend the Gauss hypergeometric matrix function in the following form: C. is matrix power series is seen to converge when |z| < 1. Let A and B be positive stable matrices in Cr×r; the Jacobi matrix polynomial (JMP) P(nA,B)(z) is defined by

Generating Functions of the EGHMF

Integral Representations for the EGHMF

Conflicts of Interest