A study on certain properties of generalized M-special matrix functions and its applications

The aim of this work is to introduce generalized M-gamma and M-beta matrix functions based on the generalized M-series with matrix arguments. By employing the beta function, we define several important classes of hypergeometric matrix functions, including the generalized M-Gauss, M-confluent, M-Appell, and M-Lauricella hypergeometric matrix functions. Various particular cases are examined to illustrate how these newly defined functions extend and unify several existing results in matrix analysis. Furthermore, we establish fundamental properties such as integral representations and derivative formulas. As an application, the Laplace transforms of the M-Gauss and M-confluent hypergeometric matrix functions are derived, and their significance in fractional calculus is explored. The proposed framework offers a powerful analytical tool for solving fractional differential equations and significantly broadens the scope of matrix-valued special function theory.