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

SIGMA 14 (2018), 072, 24 pages      arXiv:1802.09190      https://doi.org/10.3842/SIGMA.2018.072
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications (OPSFA14)

Generalized Burchnall-Type Identities for Orthogonal Polynomials and Expansions

Mourad E.H. Ismail a, Erik Koelink b and Pablo Román c
a) University of Central Florida, Orlando, Florida 32816, USA
b) IMAPP, Radboud Universiteit, PO Box 9010, 6500GL Nijmegen, The Netherlands
c) CIEM, FaMAF, Universidad Nacional de Córdoba, Medina Allende s/n Ciudad Universitaria, Córdoba, Argentina

Received February 27, 2018, in final form July 11, 2018; Published online July 17, 2018

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.

Key words: orthogonal polynomials; Askey scheme and its $q$-analogue; expansion formulas; Toda lattice.

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