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arXiv · 1905.10420

Convolution identities for Dunkl orthogonal polynomials from the $\mathfrak{osp}(1|2)$ Lie superalgebra

Erik Koelink·Jean-Michel Lemay·Luc Vinet
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Abstract

New convolution identities for orthogonal polynomials belonging to the $q=-1$ analog of the Askey-scheme are obtained. A specialization of the Chihara polynomials will play a central role as the eigenfunctions of a special element of the Lie superalgebra $\mathfrak{osp}(1|2)$ in the positive discrete series representation. Using the Clebsch-Gordan coefficients, a convolution identity for the Specialized Chihara, the dual -1 Hahn and the Big -1 Jacobi polynomials is found. Using the Racah coefficients, a convolution identity for the Big -1 Jacobi and the Bannai-Ito polynomials is found. Finally, these results are applied to construct a bilinear generating function for the Big -1 Jacobi polynomials.

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Erik Koelink, Jean-Michel Lemay, Luc Vinet. 2019-05-24. Convolution identities for Dunkl orthogonal polynomials from the $\mathfrak{osp}(1|2)$ Lie superalgebra. https://doi.org/10.1063/1.5111344

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