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

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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.

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Authors and Affiliations

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

    Antonio J. Duran & Pedro Lopez-Rodriguez

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  1. Antonio J. Duran
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  2. Pedro Lopez-Rodriguez
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Correspondence to Antonio J. Duran or Pedro Lopez-Rodriguez.

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Duran, A., Lopez-Rodriguez, P. Structural Formulas for Orthogonal Matrix Polynomials Satisfying Second-Order Differential Equations, II. Constr Approx 26, 29–47 (2007). https://doi.org/10.1007/s00365-006-0637-4

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  • DOI: https://doi.org/10.1007/s00365-006-0637-4

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