A Hahn-Type Characterization of Generalized Hermite Polynomials Through a Dunkl-Based Raising Operator

Khalid Ali Alanezy () and Jihad Souissi
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Khalid Ali Alanezy: Department of Mathematics, King Fahd University of Petroleum & Minerals (KFUPM), Dhahran 31261, Saudi Arabia
Jihad Souissi: Department of Mathematics, Faculty of Sciences, University of Gabes, Gabes 6072, Tunisia

Mathematics, 2025, vol. 13, issue 21, 1-14

Abstract: In this paper, we study Hahn’s problem with respect to a Dunkl-perturbed raising operator. More precisely, we prove that, up to a dilation, the generalized Hermite polynomials are the only T μ , α -classical symmetric orthogonal polynomials, where T μ , α = T μ + α t I , α ∈ C ∖ { 0 } and I denotes the identity operator on the space of polynomials with complex coefficients. The argument uses an operator product rule for T μ , duality for the associated functionals, and a symmetry-enforced identification together with matching three-term recurrences. The result provides an operator-theoretic Hahn-type characterization that complements semiclassical Pearson-equation descriptions and clarifies the effect of the raising perturbation α t I .

Keywords: orthogonal polynomials; symmetric forms; Dunkl operator; semiclassical polynomials (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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