A Bochner Theorem for Dunkl Polynomials

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 $q=-1$.

References in corpus (1)

• A "missing" family of classical orthogonal polynomials

Cited by in corpus (14)

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• Supersymmetric Quantum Mechanics with Reflections
• The Bannai-Ito algebra and a superintegrable system with reflections on the 2-sphere
• From $s{l}_q(2)$ to a Parabosonic Hopf Algebra
• Dunkl shift operators and Bannai-Ito polynomials
• The algebra of dual -1 Hahn polynomials and the Clebsch-Gordan problem of sl_{-1}(2)
• A "Continuous" Limit of the Complementary Bannai-Ito Polynomials: Chihara Polynomials
• Dunkl-Supersymmetric Orthogonal functions associated with classical orthogonal polynomials
• An infinite family of superintegrable Hamiltonians with reflection in the plane
• Jordan algebras and orthogonal polynomials
• Bivariate Bannai-Ito polynomials
• CMV matrices and little and big -1 Jacobi polynomials
• A superintegrable model with reflections on $S^3$ and the rank two Bannai-Ito algebra
• Convolution identities for Dunkl orthogonal polynomials from the $\mathfrak{osp}(1|2)$ Lie superalgebra