The halves of Delannoy matrix and Chung-Feller properties of the m-Schröder paths

Abstract

We study the (m,r,s)-halves of the Delannoy matrix and Delannoy matrix of the second kind. We find expression for the m-central coefficients and the (m,r,s)-halves for these Delannoy matrices in terms of the generating function of the (m+1)-Schröder numbers. We obtain from these results some new Chung-Feller properties for the m-Schröder paths, and we also give a bijective proof for one of the results. As an additional application we obtain some identities related to the m-Schröder numbers.