An infinite dimensional umbral calculus

Abstract

The aim of this paper is to develop foundations of umbral calculus on the space D′ of distributions on Rd, which leads to a general theory of Sheffer polynomial sequences on D′. We define a sequence of monic polynomials on D′, a polynomial sequence of binomial type, and a Sheffer sequence. We present equivalent conditions for a sequence of monic polynomials on D′ to be of binomial type or a Sheffer sequence, respectively. We also construct a lifting of a sequence of monic polynomials on R of binomial type to a polynomial sequence of binomial type on D′, and a lifting of a Sheffer sequence on R to a Sheffer sequence on D′. Examples of lifted polynomial sequences include the falling and rising factorials on D′, Abel, Hermite, Charlier, and Laguerre polynomials on D′. Some of these polynomials have already appeared in different branches of infinite dimensional (stochastic) analysis and played there a fundamental role.