A new characterization and a Rodrigues formula for generalized Hermite orthogonal polynomials

Abstract

In this paper, we consider the raising operator Rξ = ξ Tμ + xI, ξ ̸= 0, where Tμ and I are the Dunkl operator and the identity operator respec- tively. Our purpose is to determine all monic orthogonal polynomials se- quences {Pn(x)}n≥0 such that the sequence of polynomials {(Rξ Pn)(x)}n≥0 is also orthogonal. We prove that the only sequence of polynomials sat- isfying this condition is, up to a dilation, the generalized Hermite poly- nomial sequence. Then, we explore our result to deduce a Rodrigues formula for the generalized Hermite polynomials sequence.