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 (Formula presented.) -classical symmetric orthogonal polynomials, where (Formula presented.), (Formula presented.) and (Formula presented.) denotes the identity operator on the space of polynomials with complex coefficients. The argument uses an operator product rule for (Formula presented.), 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 (Formula presented.).