Finkelshtein, Dmitri L.Kondratiev, Yuri G.Lytvynov, EugeneOliveira, Maria João2019-07-012019-07-012019-06-16Finkelshtein, Dmitri; [et al.] - An infinite dimensional umbral calculus. "Journal of. Functional Analysis" [Em linha]. ISSN 0022-1236. Vol. 276, nº 12 (2019), p. 3714-37660022-1236http://hdl.handle.net/10400.2/8369The 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.engGenerating functionPolynomial sequence on D'Polynomial sequence of binomial type on D'Sheffer sequence on D'Shift-invarianceUmbral calculus on D'journal article10.1016/j.jfa.2019.03.006