Umbral calculus over a vector space

Abstract

Let $V$ be a vector space over $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$. We develop a basis-free umbral calculus over $V$. We define the vector space of polynomials over $V$, and polynomial sequences in it. We discuss shift-invariant operators acting in polynomials over $V$. We define polynomial sequences of binomial type and Sheffer sequences over $V$. We provide equivalent characterizations of these polynomial sequences. We prove two recurrence formulas for Sheffer sequences. With each Sheffer sequence, we associate a linear operator acting in polynomials over $V$, which we call a Sheffer operator. We prove that the set of Sheffer operators is a group for the usual product of linear operators, which is isomorphic to the Riordan group of pairs of formal tensor power series in a variable from $V$. Under the assumption that $V$ is an algebra, we lift every Sheffer sequence over $\mathbb{F}$ to a Sheffer sequence over $V$. We provide examples of such lifting.