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

SIGMA 7 (2011), 020, 9 pages      arXiv:1011.1457      https://doi.org/10.3842/SIGMA.2011.020
Contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S4)”

A Bochner Theorem for Dunkl Polynomials

Luc Vinet a and Alexei Zhedanov b
a) Centre de recherches mathématiques Universite de Montréal, P.O. Box 6128, Centre-ville Station, Montréal (Québec), H3C 3J7 Canada
b) Donetsk Institute for Physics and Technology, Donetsk 83114, Ukraine

Received November 30, 2010, in final form February 25, 2011; Published online February 27, 2011

Abstract
We establish an analogue of the Bochner theorem for first order operators of Dunkl type, that is we classify all such operators having polynomial solutions. Under natural conditions it is seen that the only families of orthogonal polynomials in this category are limits of little and big q-Jacobi polynomials as q=−1.

Key words: classical orthogonal polynomials; Dunkl operators; Jacobi polynomials; little q-Jacobi polynomials; big q-Jacobi polynomials.

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