Hermite Matrix-Valued Functions Associated to Matrix Differential Equations

Some sequences of matrix polynomials have been introduced recently as solutions of certain second-order differential equations, which can be seen as appropriate generalizations, to the matrix setting, of classical orthogonal polynomials. In this paper, we consider families (in a complex parameter) of matrix-valued special functions of Hermite type, which arise as natural extensions of the aforementioned matrix polynomials of the same type. We show that such families are solutions of corresponding differential equations and enjoy several structural properties. In particular, they satisfy a Rodrigues formula expressed in terms of the Weyl fractional calculus. We also show that, unlike the scalar case, a second-order differential operator having such a family as a set of joint eigenfunctions need not be unique.