Riordan Arrays, Sheffer Polynomials and the Stirling Transform

Abstract

This study brings together a connection among the Sheffer polynomial sequences, the Riordan arrays and the Stirling numbers. Beside the algebraic view, a differential equation that gives rise to Sheffer sequences is discussed; serving as a differential approach to the study of Sheffer polynomials. Numbers generated by the function eet􀀀1 := eGt have already been shown to satisfy the congruence relation Gn+p _ Gn + Gn+1(mod p) for any p prime. We derive a congruence relation for the Touchard polynomial sequence and the Stirling numbers of the second kind analogous to the known congruence relation for the Gn. Also, some Riordan arrays with generating functions related to eet􀀀1 for polynomial sequences such as the Touchard polynomial, Toscano polynomial, Charlier polynomial and the Poisson - Charlier polynomial are discussed, and their corresponding inverses constructed.