Combinatorial identities involving the central coefficients of a Sheffer matrix

Given m ∈ N, m ≥ 1, and a Sheffer matrix S = [s_{n,k}]_{n,k}≥0, we obtain the exponential generating series for the coefficients binomial(a+(m+1)n,a+mn)^{−1}*s_{a+(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.