Combinatorial identities involving the central coefficients of a Sheffer matrix

Abstract

Given m ? N, m ? 1, and a Sheffer matrix S = [sn,k]n,k?0, we obtain the exponential generating series for the coefficients (a+(m+1)n a+mn)-1 sa+(m+1)n,a+mn. Then, by using this series, we obtain two general combinatorial identities, and their specialization to r-Stirling, r-Lah and r-idempotent numbers. In particular, using this approach, we recover two well known binomial identities, namely Gould's identity and Hagen-Rothe's identity. Moreover, we generalize these results obtaining an exchange identity for a cross sequence (or for two Sheffer sequences) and an Abel-like identity for a cross sequence (or for an s-Appell sequence). We also obtain some new Sheffer matrices.