The Riordan Arrays And Matrix Identities

Combinatorial sequences and combinatorial identities are important parts of combinatorics and are deeply connected with many discrete problems. In our thesis, we concern on certain sequences and identities, and get some results involving them. The content of this thesis can be summarized as follows:In Chapter1, we present the current situation of some combinatorial sequences and iden-tities, and the development of Riordan array theorem as well.In Chapter2, we introduce the definitions and properties of Dyck path and Motzkin path. Based on the concept of Dyck path, we give the definition of b-ary path and its enumeration. We give a brief introduction of Riordan array theorem.In Chapter3, it is devoted to counting problem of return points of n-ary path. Let wb(n, k) denote the number of b-ary paths of length (b+1)n that contain exactly k return points. We get explicit expression for wb(n, k) by using the generating function and Lagrange inversion formula. Furthermore, we construct a Riordan array in which we put wb(n, k) as an element of the n-th row and the k-th column.In Chapter4, we obtain several matrix identities by using the Riordan array theorem. For a special case, the Chebyshev polynomials of the second kind satisfy our identity. Based on special lattice paths and number sequences, we also present combinatorial proofs for two matrix identities related to odd terms of Fibonacci numbers.