Generalized Burchnall-Type Identities For Orthogonal Polynomials And Expansions

Keywords

Askey scheme and its q-analogue; Expansion formulas; Orthogonal polynomials; Toda lattice

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 and 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.

Publication Date

7-17-2018

Publication Title

Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

Volume

14

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.3842/SIGMA.2018.072

Socpus ID

85050356045 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/85050356045

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