A general matrix series inversion pair and associated polynomials

Abstract

In the present work, a pair of general inverse matrix series relations is established, and thereby a general class of matrix polynomials is introduced. This class generalizes the extended Jacobi polynomials and their particular cases such as the polynomials of Brafman, Jacobi, Chebyshev, and Legendre. It is further shown that this pair also gives rise to the matrix forms of the Wilson polynomials and the Racah polynomials. For these polynomials, the generating matrix function relations as well as the matrix summation formulas are deduced from their respective inverse pairs. Certain inverse pairs belonging to the Gould classes and the Legendre-Chebyshev classes due to John Riordan [An Introduction to Combinatorial Identities, Wiley, 1968] are also extended to matrix forms.

An Extended Version of the _r+1<em>R</em>_s;k(B;C; <em>z</em>) Matrix Function

“…By repeating the above process that µ times, we get (32). 22)-( 24), we get the results for the generalized hypergeometric matrix functions [34].…”

“…where E A B,C (z) is the three parameter Mittag-Leffler matrix function given by Sanjhira et al [32] Case 2. On setting k = 1, (20) reduces to r+1 R s,1 (A, P 1 , P 2 , .…”

“…For the past four decades or so, in both mathematics and science, certain special matrix functions are crucial. Jódar and Sastre [11], Jódar and Cortés [12][13][14] researched the matrix analogues of the gamma, beta, and Gauss hypergeometric functions, which provided the basis for the special matrix functions Bakhet et al [3], Çekim et al [4], Gezer and Kaanoglu [9].The extended work of r+1 R s (P, Q, z) matrix functions is examined in some detail in Sanjhira and Dave [30], Sanjhira and Dwivedi [31], Sanjhira et al [32], Shehata [34][35][36], Shehata et al [37], Varma et al [38] for examples of several polynomials that have been introduced and investigated from a matrix perspective. The generalization of the r+1 R s (P, Q, z) function presented here was motivated by our investigations [31,37] of the properties of a class of polynomials which characterize is itself an interesting subject, the subsequent Disclaimer/Publisher's Note: The statements, opinions, and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s).…”

Some Relations on the rRs(P,Q,z) Matrix Function

“…where n, k ∈ N and Z Q−I n (z; k) are the Konhauser matrix polynomials [16,[44][45][46][47][48] of degree n in z k .…”

“…where E P (zt) is a Mittag-Leffler matrix function. By using the relation between the Mittag-Leffler matrix function E P (zt) and the generalized Wright matrix function 2 Ψ 2 [45], we find…”