Closed form expressions for Appell polynomials

Abstract

We show that any Appell sequence can be written in closed form as a forward difference transformation of the identity. Such transformations are actually multipliers in the abelian group of the Appell polynomials endowed with the operation of binomial convolution. As a consequence, we obtain explicit expressions for higher order convolution identities referring to various kinds of Appell polynomials in terms of the Stirling numbers. Applications of the preceding results to generalized Bernoulli and Apostol-Euler polynomials of real order are discussed in detail.

Appell and Sheffer sequences: on their characterizations through functionals and examples

“…the degree of a polynomial by k. Remark 4.3. The previous proposition was recently studied in [4] for Q = ∆ 1 = ∆, the difference operator of step one. Let us recall that for each h ∈ C * , the difference operator…”

“…by using Proposition 4.2. In fact, in this case B(t) = log(1 + t) and the previous formulas follow from (19) since ((e t − 1)/t) k • B(t) = log(1 + t) k /t k , c.f., [4,Theorem 5].…”

“…Our goal in this work is to obtain a new simple recursion for Appell sequences (Theorem 4.1), their expansions in terms of an arbitrary delta operator (Proposition 4.2) -as presented in [4] for difference operators-, and several examples. Our exposition is based on Umbral Calculus, briefly recalled in Sections 2 and 3.…”