Further results on the generalised hypergeometric matrix functions

Abstract

In recent years, various results of the special matrix functions have been established in many papers. In the present paper, we have developed certain properties involving generalised hypergeometric matrix functions, such as, integral representations and reduction formulae. Also, open problems concerning the generalised hypergeometric matrix functions are stated.

A Note on the Appell Hypergeometric Matrix Function F2

“…Substituting t � (1 − z)u into (22), we obtain formula (18). Now, connections with the Gauss hypergeometric matrix function is considered by the following theorem: □ Theorem 3.…”

“…On the other hand, many authors [18][19][20][21][22][23][24][25] generalized the hypergeometric series F(α, β, c; z) by extending parameters α, β, and c to square matrices A, B, and C in the complex space C d×d . Recently, the extension of the classical Appell hypergeometric functions F s , s � {1, 2, 3, 4}, to the Appell hypergeometric matrix functions has been a subject of intensive studies [26][27][28][29][30].…”

On the Fractional Order Rodrigues Formula for the Shifted Legendre-Type Matrix Polynomials

“…In the past few years, matrix functions and polynomials defined in terms of power matrix series [18][19][20][21] attracted the attention to new and exciting problems, specifically that involve the generalized hypergeometric matrix functions and future directions. It is worth mentioning the recent work of Shang [22] who uncovered some of the information of topological properties hidden in the smaller eigenvalues of A in matrix functions of the form f (A) = ∑ ∞ k=0 c k A k .…”

“…Recently, many authors (see, e.g., [18][19][20][21]) proposed the extension of the classical polynomials to the matrix framework. The operational matrix of fractional derivatives has been studied for various types of orthogonal polynomials, including Chebyshev polynomials [27].…”

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

“…Applications of special matrix functions also grow and become active areas in the recent literature including statistics, Lie groups theory, and differential equations (see, e.g., [1][2][3][4] and elsewhere). 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].…”