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arXiv · 1802.09190

Generalized Burchnall-Type Identities for Orthogonal Polynomials and Expansions

Mourad E. H. Ismail·Erik Koelink·Pablo Román
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Abstract

Burchnall's method to invert the Feldheim-Watson linearization formula for the Hermite polynomials is extended to all polynomial families in the Askey-scheme and its $q$-analogue. The resulting expansion formulas are made explicit for several families corresponding to measures with infinite support, including the Wilson and Askey-Wilson polynomials. An integrated version gives the possibility to give alternate expression for orthogonal polynomials with respect to a modified weight. This gives expansions for polynomials, such as Hermite, Laguerre, Meixner, Charlier, Meixner-Pollaczek and big $q$-Jacobi polynomials and big $q$-Laguerre polynomials. We show that one can find expansions for the orthogonal polynomials corresponding to the Toda-modification of the weight for the classical polynomials that correspond to known explicit solutions for the Toda lattice, i.e., for Hermite, Laguerre, Charlier, Meixner, Meixner-Pollaczek and Krawtchouk polynomials.

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Mourad E. H. Ismail, Erik Koelink, Pablo Román. 2018-02-26. Generalized Burchnall-Type Identities for Orthogonal Polynomials and Expansions. https://doi.org/10.3842/sigma.2018.072

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