Binomial convolution and transformations of Appell polynomials

Abstract

We show that the set A of Appell sequences is an abelian group under the binomial convolution. This is essentially equivalent to other approaches considered in the literature, in particular, the determinantal approach. We also define a scale transformation Tα:A→A, which is an isomorphism, and a transformation RY:A→A based on expectations with respect to a random variable Y. Using these tools, we give explicit formulas for the determinantal form of Appell sequences, as well as for the formula of representation of powers and the Srivastava–Pintér addition theorem. Various illustrative examples, mainly referring to various generalizations of Bernoulli and Euler polynomials, are discussed in detail.