Refined Chung–Feller theorems for lattice paths

In this paper we prove a strengthening of the classical Chung–Feller theorem and a weighted version for Schröder paths. Both results are proved by refined bijections which are developed from the study of Taylor expansions of generating functions. By the same technique, we establish variants of the bijections for Catalan paths of order d and certain families of Motzkin paths. Moreover, we obtain a neat formula for enumerating Schröder paths with flaws.