Abstract
We develop structural formulas satisfied by some families of orthogonal matrix polynomials of size $2\times 2$ satisfying second-order differential equations with polynomial coefficients. We consider here two one-parametric families of weight matrices, namely \[ H_{a,1}(t)\;=\;e^{-t^2} \left( \begin{array}{@{}cc@{}} 1+\vert a\vert ^2t^2 & at \\ \bar at & 1 \end{array} \right) \quad {\rm and} \quad H_{a,2}(t)\;=\;e^{-t^2} \left( \begin{array} {@{}cc@{}} 1+\vert a\vert ^2t^4 & at^2 \\ \bar at^2 & 1 \end{array} \right), \] $a\in \mbox{\bf C} $ and $t\in \mbox{\bf R} $, and their corresponding orthogonal polynomials.
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Durán, A., Grünbaum, F. Structural Formulas for Orthogonal Matrix Polynomials Satisfying Second-Order Differential Equations, I. Constr Approx 22, 255–271 (2005). https://doi.org/10.1007/s00365-004-0577-2
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DOI: https://doi.org/10.1007/s00365-004-0577-2