Some Results on the Extended Hypergeometric Matrix Functions and Related Functions

Mohamed Abdalla,Sahar Ahmed Idris,Ibrahim Mekawy,Hemant Kumar Nashine

doi:10.1155/2021/2046726

Mohamed Abdalla, Sahar Ahmed Idris + Show 2 more

Open Access

https://doi.org/10.1155/2021/2046726

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Abstract

In this article, we discuss certain properties for generalized gamma and Euler’s beta matrix functions and the generalized hypergeometric matrix functions. The current results for these functions include integral representations, transformation formula, recurrence relations, and integral transforms.

Highlights

Matrix generalizations of some known classical special functions are important both from the theoretical and applied point of view. ese new extensions have proved to be very useful in various fields such as physics, engineering, statistics, actuarial sciences, life testing, and telecommunications
We introduced new extensions of the Gauss hypergeometric matrix function and confluent hypergeometric matrix function. en, we discussed certain properties of these extended matrix functions such as the integral representations, transformation formulae, recurrence relations, and integral transforms
Some interesting special cases of our main results are archived

Summary

Introduction

Matrix generalizations of some known classical special functions are important both from the theoretical and applied point of view (see, for example, [1,2,3,4,5,6,7,8,9,10]). ese new extensions have proved to be very useful in various fields such as physics, engineering, statistics, actuarial sciences, life testing, and telecommunications. Let A be a positive stable matrix in Cr×r; the gamma matrix function in [11, 12] is defined by. In the recent paper [14], for any arbitrary parameter p with Re(p) > 0, the matrix generalizations of gamma and Euler’s beta functions are given as follows: Γ(X; p). Respectively, where A, B, X, and Y are positive stable matrices in Cr×r and p is any arbitrary parameter with Re(p) > 0 These are matrix versions of gamma and beta functions [23]. E case of A B in (12) and (13) gives us generalizations of gamma and Euler’s beta matrix functions defined by (10) and (11), respectively.

Properties of Generalizations of Gamma and Beta Matrix Functions

Properties of the GGHMF and GCHMF