# An infinite dimensional umbral calculus

## TL;DR

This paper develops an infinite-dimensional umbral calculus framework on distribution spaces, extending classical polynomial sequences to this setting and providing new tools for infinite-dimensional analysis.

## Contribution

It introduces a foundational theory of umbral calculus on distribution spaces, including definitions, characterizations, and liftings of polynomial sequences of binomial and Sheffer types.

## Key findings

- Constructed liftings of polynomial sequences from  to '
- Provided conditions for binomial and Sheffer sequences on distribution spaces
- Examples include lifted factorials, Abel, Hermite, Charlier, and Laguerre polynomials

## Abstract

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