Generating orthogonal matrix polynomials satisfying second order differential equations from a trio of triangular matrices |
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Authors: | Antonio J. Dur n |
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Affiliation: | aDepartamento de Análisis Matemático, Universidad de Sevilla, Apdo (P. O. BOX) 1160, 41080 Sevilla, Spain |
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Abstract: | The method developed in [A.J. Durán, F.A. Grünbaum, Orthogonal matrix polynomials satisfying second order differential equations, Int. Math. Res. Not. 10 (2004) 461–484] led us to consider matrix polynomials that are orthogonal with respect to weight matrices W(t) of the form , , and (1−t)α(1+t)βT(t)T*(t), with T satisfying T′=(2Bt+A)T, T(0)=I, T′=(A+B/t)T, T(1)=I, and T′(t)=(−A/(1−t)+B/(1+t))T, T(0)=I, respectively. Here A and B are in general two non-commuting matrices. We are interested in sequences of orthogonal polynomials (Pn)n which also satisfy a second order differential equation with differential coefficients that are matrix polynomials F2, F1 and F0 (independent of n) of degrees not bigger than 2, 1 and 0 respectively. To proceed further and find situations where these second order differential equations hold, we only dealt with the case when one of the matrices A or B vanishes.The purpose of this paper is to show a method which allows us to deal with the case when A, B and F0 are simultaneously triangularizable (but without making any commutativity assumption). |
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Keywords: | Orthogonal matrix polynomials Second order differential equations |
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