### 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

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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