In this paper, we introduce Shively’s Pseudo Laguerre type matrix polynomials and obtain some properties, including the hypergeometric matrix function representation, the \({}_pR_q(\mathcal {P},\mathcal {Q};z)\) matrix function representation, and the generating matrix functions. Furthermore, we establish the composition of generalized fractional calculus operators with the Shively’s Pseudo Laguerre type matrix polynomials, and some special cases have also been discussed. The proposed generating function provides advanced tools for analyzing and solving matrix-based systems encountered in quantum mechanics, control theory, and numerical analysis.
