On the matrix function $_pR_q(A, B; z)$ and its fractional calculus properties

TL;DR

This paper introduces a new matrix function $_pR_q(A, B; z)$, explores its properties, convergence, and fractional calculus aspects, and discusses special cases like matrix Mittag-Leffler functions and matrix polynomials.

Contribution

It presents the first comprehensive study of the matrix function $_pR_q(A, B; z)$, including its properties, relations, and fractional calculus applications.

Findings

Derived differential formulas and integral representations.

Established convergence criteria for the matrix function.

Explored special cases such as matrix Mittag-Leffler functions.

Abstract

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.