Sums of Products of Cauchy Numbers, Including Poly-Cauchy Numbers

We investigate sums of products of Cauchy numbers including poly-Cauchy numbers: . A relation among these sums shown in the paper and explicit expressions of sums of two and three products (the case of and that of described in the paper) are given. We also study the other three types of sums of products related to the Cauchy numbers of both kinds and the poly-Cauchy numbers of both kinds. 1. Introduction The Cauchy numbers (of the first kind) are defined by the integral of the falling factorial: (see [1, Chapter VII]). The numbers are sometimes called the Bernoulli numbers of the second kind (see e.g., [2, 3]). Such numbers have been studied by several authors [4–8] because they are related to various special combinatorial numbers, including Stirling numbers of both kinds, Bernoulli numbers, and harmonic numbers. It is interesting to see that the Cauchy numbers of the first kind have the similar properties and expressions to the Bernoulli numbers . For example, the generating function of the Cauchy numbers of the first kind is expressed in terms of the logarithmic function: (see [1, 6]), and the generating function of Bernoulli numbers is expressed in terms of the exponential function: (see [1]) or (see [9]). In addition, Cauchy numbers of the first kind can be written explicitly as (see [1, Chapter VII], [6, page 1908]), where are the (unsigned) Stirling numbers of the first kind, arising as coefficients of the rising factorial (see e.g., [10]). Bernoulli numbers (in the latter definition) can be also written explicitly as where are the Stirling numbers of the second kind, determined by (see, e.g., [10]). Recently, Liu et al. [5] established some recurrence relations about Cauchy numbers of the first kind as analogous results about Bernoulli numbers by Agoh and Dilcher [11]. In 1997 Kaneko [9] introduced the poly-Bernoulli numbers ( , ) by the generating function where is the th polylogarithm function. When , is the classical Bernoulli number with . On the other hand, the author [12] introduced the poly-Cauchy numbers (of the first kind) as a generalization of the Cauchy numbers and an analogue of the poly-Bernoulli numbers by the following: In addition, the generating function of poly-Cauchy numbers is given by where is the th polylogarithm factorial function, which is also introduced by the author [12, 13]. If , then is the classical Cauchy number. The following identity on sums of two products of Bernoulli numbers is known as Euler’s formula: The corresponding formula for Cauchy numbers was discovered in [8]: In this paper, we shall give more