Finite Orthogonal M Matrix Polynomials

In this study, we aim to construct a finite set of orthogonal matrix polynomials for the first time, along with their finite orthogonality, matrix differential equation, Rodrigues' formula, several recurrence relations including three-term relation, forward and backward shift operators, generating functions, integral representation and their relation with Jacobi matrix polynomials. Thus, the concept of "finite", which is used to impose parametric constraints for orthogonal polynomials, is transferred to the theory of matrix polynomials for the first time in the literature. Moreover, this family reduces to the finite orthogonal M polynomials in the scalar case when the degree is 1, thereby providing a matrix generalization of finite orthogonal M polynomials in one variable.