The main objective of the present paper is to introduce and study the
function $_pR_q(A, B; z)$ with matrix parameters and investigate the
convergence of this matrix function. The contiguous matrix function relations,
differential formulas and the integral representation for the matrix function
$_pR_q(A, B; z)$ are derived. Certain properties of the matrix function
$_pR_q(A, B; z)$ have also been studied from fractional calculus point of view.
Finally, we emphasize on the special cases namely the generalized matrix
$M$-series, the Mittag-Leffler matrix function and its generalizations and some
matrix polynomials.
We propose a matrix analogue of a general inverse series relation with an objective to introduce the generalized Humbert matrix polynomial, Wilson matrix polynomial, and the Rach matrix polynomial together with their inverse series representations. The matrix polynomials of Kiney, Pincherle, Gegenbauer, Hahn, Meixner-Pollaczek etc. occur as the special cases. It is also shown that the general inverse matrix pair provides the extension to several inverse pairs due to John Riordan [An Introduction to Combinatorial Identities, Wiley, 1968].
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.
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