On Crofton–Glaisher type relations and derivation of generating functions for Hermite polynomials including the multi-index case

Abstract The Glaisher rule is an operational identity involving the action of an exponential operator containing the second-order derivatives acting on an exponential function. We use the Crofton and monomiality formalism to derive generalized forms to the multi-dimensional case and show its usefulness in the derivation of old and new forms of generating functions for a wealth of Hermite polynomials families.