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Structure relations for orthogonal polynomials. (English) Zbl 1166.42014

This paper is devoted to study how to derive structure relations for general orthogonal polynomials, that is, to find operators whose action on \(p_n\) is a combination of \(p_n\) and \(p_{n+1}\) with variable coefficients.
The results of T. H. Koornwinder [J. Comput. Appl. Math. 207, No. 2, 214–226 (2007; Zbl 1120.33018)] about structure relations with constant coefficients are generalized to differential operators and polynomials orthogonal with respect to some exponential weight functions. The aproach given here uses an alternative to the Sturm-Liouville property for the orthogonal polynomials. Moreover, the results of G. Bangerezako [J. Comput. Appl. Math. 107, No. 2, 219–232 (1999; Zbl 0933.39042)] are also extended to a more general class of weight functions.
Finally, the orthogonal polynomials with respect to a generalized Jacobi weight evolved under a Toda-type modification are also treated.

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33C47 Other special orthogonal polynomials and functions
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
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