Structure relations for orthogonal polynomials
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 \textit{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 \textit{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.
scientific article; zbMATH DE number 5541996