Combinatorial identities involving the central coefficients of a Sheffer matrix

Abstract

Given m \uf0ce N, m ≥ 1, and a Sheffer matrix S = [sn,k]n,k≥0, we obtain the \n exponential generating series for the coefficients (a+(m+1)n a+mn)-1 \n sa+(m+1)n,a+mn. Then, by using this series, we obtain two general \n combinatorial identities, and their specialization to r-Stirling, r-Lah and \n r-idempotent numbers. In particular, using this approach, we recover two \n well known binomial identities, namely Gould s identity and Hagen-Rothe s \n identity. Moreover, we generalize these results obtaining an exchange \n identity for a cross sequence (or for two Sheffer sequences) and an Abel-like \n identity for a cross sequence (or for an s-Appell sequence). We also obtain \n some new Sheffer matrices.