Taylor expansions for the m-Catalan numbers

By a Taylor expansion of a generating function, we mean that the remainder of the expansion is a functional of the generating function itself. In this paper, we consider the Taylor expansion for the generating function Bm(t) of them-Catalan numbers. In order to give combinatorial interpretations of the coefficients of these expansions, we study a new collection of partial Grand Dyck paths, that is, (i, j)-balance m-Dyck paths, and we obtain some new Chung-Feller type results.