The Product Formula and Convolution Structure Associated with the Generalized Hermite Polynomials

Abstract

Here the product formula for the generalized and suitably normalized Hermite polynomials with parameter μ ≥ 0 will be explicitly established. Its measure turns out to be absolutely continuous and supported on two disjoint intervals lying symmetrically on the real line, provided that μ > 0. In the limit case μ = 0, which is associated with the classical Hermite polynomials, four additional point masses occur at the endpoints of the two intervals. As an application, the product formula is used to introduce a generalized translation operator and a corresponding convolution product on appropriately weighted Lebesgue spaces. To this end, norm estimates of the translation operator from above and below are presented. For any μ ≥ 1 2 , this gives rise to a quasi-positive convolution algebra.