Structural Formulas for Orthogonal Matrix Polynomials Satisfying Second-Order Differential Equations, I

Abstract

We develop structural formulas satisfied by some families of orthogonal matrix polynomials of size 2 × 2 satisfying second-order differential equations with polynomial coefficients. We consider here three one-parametric families of weight matrices, namely, $$A_{a,1}(t)=t^{\alpha } e^{-t}\left(\begin{array}{@{}cc@{}} 1+\vert a\vert ^2t^2 &at \\ \bar at & 1 \end{array}\right), $$ $$A_{a,2}(t)=t^{\alpha } e^{-t}\left(\begin{array}{@{}cc@{}} t^2+(t-1)^2\vert a\vert ^2 &a(t-1) \\ \bar a(t-1) & 1 \end{array}\right),\ \ a\in {\bf C} \mbox{and} t\in (0,+\infty), $$ and $$A_{a,3}(t)=(1-t)^{\alpha }(1+t)^\beta\left(\begin{array}{@{}cc@{}} (1+t)^2+t^2\vert a\vert ^2 &at \\ \bar at & 1 \end{array}\right),\ \ a\in {\bf C} \mbox{and} t\in (-1,1), $$ and their corresponding orthogonal polynomials. We also show that the orthogonal polynomials with respect to the second family are eigenfunctions of two linearly independent second-order differential operators.