Riordan Arrays And The Generalization Of Chung-Feller Theorem

In this paper,some Riordan matrices of the generalized Pell walks are obtained by utilizing the Riordan array's A-matrix,whose row sum will be satisfied the recursion involved the generalized Pell numbers.In this way,the combinatorial interpretation of the generalized Pell numbers is acquired.Furthermore,some Riordan matrices of the generalized Pell walks which have no steps below the line = are obtained.It is proved that the elements of the first column of the Riordan matrix composed of the restricted lattice path and the elements on the center line of the corresponding unrestricted Riordan matrix satisfy the property of the ChungFeller Theorem,and the combinatorial proof is given.By using the same method,it is proved that the 3-Dyck path satisfies the property of the Chung-Feller Theorem.Finally,the bijection between the 3-Dyck path and the completely ternary tree is established.Then the property is extended to the complete k-ary trees.