Riordan Arrays and Difference Equations of Subdiagonal Lattice Paths

We study lattice paths by combinatorial methods on the positive lattice. We give some identity that produces the functional equationsand generating functions to counting the lattice paths on or below the main diagonal.Also, we consider the subdiagonal lattice paths in relation to lower triangular arrays.This presents a Riordan array in conjunction with the columns of the matrix of the coefficients ofcertain formal power series by implying an infinite lower triangular matrix $ F=(f_{x,y})_{x,y\geqslant 0} $ .We derive new combinatorial interpretations in terms of restricted lattice paths for some Riordan arrays.