Enumeration of Lattice paths with infinite types of steps and the Chung-Feller property

In this paper we consider the lattice paths that go from the origin to a point on the right half of the plane with step set S = {U-i = (1 + i, 1 - i)vertical bar i >= 0} boolean OR {V-i = (i, -i)vertical bar i >= 1} such that the step U-i is assigned with weight u(i) and the step Viis assigned with weight v(i) for i >= 1 except that the step U-0 = (1, 1) is assigned with weight 1. Let G(n, k) be the set of all lattice paths ending at the point (n, 2k - n) and G(n,k) = vertical bar G(n, k)vertical bar, and let U(n, k) (resp. V(n, k)) be the set of all lattice paths (resp. nonnegative lattice paths) ending at (2n - k, k) and U-n,U-k = vertical bar U(n, k)vertical bar(resp. V-n,V-k = vertical bar V(n, k)vertical bar). We will show that (G(n, k))(n,k is an element of N), (U-n,U-k)(n,k is an element of N) and (V-n,V-k)(n,k is an element of N) are all Riordan arrays. Correlations between these Riordan arrays are studied. Consequently, a new Chung-Feller type property is obtained, and the bijective proof is provided. We also list numerous interesting examples. (C) 2021 Elsevier B.V. All rights reserved.