The halves of Delannoy matrix and Chung-Feller properties of the m-Schr?der paths

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) -Schroder numbers. We obtain from these results some new Chung -Feller properties for the m -Schroder 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 -Schroder numbers. (c) 2023 Elsevier Inc. All rights reserved.