Generalized Burchnall-Type Identities for Orthogonal Polynomials and Expansions
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 -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 -Jacobi polynomials and big -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.
Cite
@article{arxiv.1802.09190,
title = {Generalized Burchnall-Type Identities for Orthogonal Polynomials and Expansions},
author = {Mourad E. H. Ismail and Erik Koelink and Pablo Román},
journal= {arXiv preprint arXiv:1802.09190},
year = {2018}
}