New developments in Gamma and Beta matrix functions based on generalized Mittag-Leffler matrix function

In this paper, we introduce and analyze new generalizations of the Gamma and Beta matrix functions based on the six-parameter Mittag-Leffler matrix function and the generalized Gamma matrix function. These extensions provide a unified framework that encompasses several classical and modern special functions as particular cases. We investigate key analytical properties of the proposed matrix functions, including recurrence relations, integral representations, and limiting behaviors. The introduced functions are of potential interest in various applications involving fractional calculus, matrix analysis, and mathematical physics. To demonstrate the effectiveness and relevance of our approach, we also provide a numerical example along with graphical interpretations to illustrate the behavior and applicability of the proposed functions.