Riordan Array Theory And Its Application In Cauchy Numbers Research

In combinatorics research, one of the main contents is looking for the way of proving the identity, especially the identity including the special combinatorial numbers. Based on the theory of the Riordan array and the (exponential) partial Bell polynomials, this paper gets some properties of the generalized Cauchy numbers and a few important identities which include special combinatorial numbers.The main results are as follows:1. The introduction to the recent development of the Riordan array,the Cauchy numbers,Faàdi Bruno formula and the (exponential) partial Bell polynomials.2. By constantα,k andλ, this paper generalizes two kinds of the Cauchy numbers and gets three kinds of generalized Cauchy numbers (α-Cauchy numbers,k-Cauchy numbers andλ-Cauchy numbers) with their generating functions. Based on these results, this paper gets some characters about these generalized Cauchy numbers and some combinatorial identities including these generalized Cauchy numbers.3. This paper gains some combinatorial identities by using the Faàdi Bruno formula, inversion formula of Lagrange and the (exponential) partial Bell polynomials.