A symbolic handling of Sheffer polynomials

Abstract

We revisit the theory of Sheffer sequences by means of the formalism introduced in Rota and Taylor (SIAM J Math Anal 25(2):694-711, 1994) and developed in Di Nardo and Senato (Umbral nature of the Poisson random variables. Algebraic combinatorics and computer science, pp 245-256, Springer Italia, Milan, 2001, European J Combin 27(3):394-413, 2006). The advantage of this approach is twofold. First, this new syntax allows us noteworthy computational simplification and conceptual clarification in several topics involving Sheffer sequences, most of the open questions proposed in Taylor (Comput Math Appl 41:1085-1098, 2001) finds answer. Second, most of the results presented can be easily implemented in a symbolic language. To get a general idea of the effectiveness of this symbolic approach, we provide a formula linking connection constants and Riordan arrays via generalized Bell polynomials, here defined. Moreover, this link allows us to smooth out many results involving Bell Polynomials and Lagrange inversion formula. © 2010 Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag.