Hobson's formula for Dunkl operators and its applications

Abstract

We generalize classical Hobson's formula concerning partial derivatives of radial functions on a Euclidean space to a formula in the Dunkl analysis. As applications we give new simple proofs of known results involving Maxwell's representation of harmonic polynomials, Bochner-Hecke identity, Pizzetti formula for spherical mean, and Rodrigues formula for Hermite polynomials.

An extension of Pizzetti's formula associated with the Dunkl operators

“…Pizzetti's formula associated with the Dunkl operators was established by [14,2]. In [21] we proved (3.5) for f ∈ P as an application of Hobson's formula associated with the Dunkl operators. (Note there is an obvious mistake of unnecessary factor (−1) n in [21, Corollary 4.5].)…”

“…Our proof is different from theirs. We also deduce the extended Pizzetti's formula from Pizzetti's formula and Hobson's formula [21].…”

“…We recall Hobson's formula associated with the Dunkl operators. Theorem 3.4 (Hobson's formula [21]).…”