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

Authors:

Antonio J. Duran Pedro Lopez-Rodriguez

Affiliation:

(1) Departamento de Analisis Matematico, Universidad de Sevilla, Apdo (P.O. Box) 1160, 41080 Sevilla, Spain

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,

$A_{a,1}(t)=t^{alpha } e^{-t}left(begin{array}{@{}cc@{}} 1+vert avert ^2t^2 &at bar at & 1 end{array}right),$

$A_{a,2}(t)=t^{alpha } e^{-t}left(begin{array}{@{}cc@{}} t^2+(t-1)^2vert avert ^2 &a(t-1) bar a(t-1) & 1 end{array}right), ain {bf C} mbox{and} tin (0,+infty),$

and

$A_{a,3}(t)=(1-t)^{alpha }(1+t)^betaleft(begin{array}{@{}cc@{}} (1+t)^2+t^2vert avert ^2 &at bar at & 1 end{array}right), ain {bf C} mbox{and} tin (-1,1),$

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.

Keywords:

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