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 R d , 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. [ABSTRACT FROM AUTHOR]