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

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 Tμ,α-classical symmetric orthogonal polynomials, where Tμ,α=Tμ+αtI, 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 Tμ, 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 αtI.