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

Takao Komatsu Graduate School of Science and Technology, Hirosaki University, Hirosaki 036-8561, Japan Correspondence should be addressed to Takao Komatsu; komatsu@cc.hirosaki-u.ac.jp Received 24 July 2012; Accepted 24 October 2012 Academic Editor: Gi Sang Cheon Copyright © 2013 Takao Komatsu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We investigate sums of products of Cauchy numbers including poly-Cauchy numbers: T m (n) = i1+⋅⋅⋅+im=n, i1 ,...,im≥0 ( n i1 ,...,im ) i1 ⋅ ⋅ ⋅ im−1 c (k) im (m ≥ 1, n ≥ 0). A relation among these sums T m (n) shown in the paper and explicit expressions of sums of two and three products (the case of m = 2 and that of m = 3 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.


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][5][6][7][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 (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 2 Journal of Discrete Mathematics (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 ( ) ( ≥ 0, ≥ 1) by the generating function where is the th polylogarithm function. When = 1, (1) = is the classical Bernoulli number with (1) 1 = 1/2. 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 = 1, then (1) = 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 analogous results by investigating a general type of sums of products of Cauchy numbers including poly-Cauchy numbers: ( 1 , . . . , whose Bernoulli version is discussed in [14]. A relation among these sums and explicit expressions of sums of two and three products are also given.

Main Results
We shall consider the sums of products of Cauchy numbers including poly-Cauchy numbers. Kamano [14] investigated the following types of sums of products: ( 1 , . . . , where Bernoulli numbers are defined by the generating function (3) and poly-Bernoulli numbers ( ) are defined by the generating function (9) and Li ( ) is the th polylogarithm function defined in (10). It is shown [14] that Consider an analogous type of sums of products of Cauchy numbers including poly-Cauchy numbers: ( 1 , . . . , Then we show the following result.

Theorem 1. For an integer and a nonnegative integer , one has
Note that the generating function of ( ) is given by Since Journal of Discrete Mathematics 3 we have Since is equal to We need the following lemma in order to prove Theorem 1.

Lemma 2. For an integer and a positive integer , one has
Proof of Lemma 2. Since we have By induction, we can show that for ≥ 1 where Thus, by using the inversion relationship (see e.g., [10,Chapter 6]), the left-hand side of the identity in the previous lemma is equal to which is the right-hand side of the desired identity.
where (1) where ( ) ( ) are poly-Cauchy polynomials of the first kind, defined by the generating function ( ) ( ) are expressed explicitly in terms of the Stirling numbers of the first kind [13, Theorem 1]: Hence, the identity (39) holds because Next, by (30) and 0 ( ) = 1 + we have Hence, Therefore, we get the identity (40). Finally, by we have Hence, Therefore, we get the identity (41).
Journal of Discrete Mathematics 5 Putting = 1 in (40), we have the following identity, which is also found in [8]. This is also an analogous formula to Euler's formula (14).

Corollary 5.
One has (see also Table 1) is also given.

Poly-Cauchy
wherê( ) is poly-Cauchy number of the second kind [12], whose generating function is given by Journal of Discrete Mathematics 7 (see [12]). In this sense, ( ) is called poly-Cauchy number of the first kind. When = 1,̂=̂( 1) is the classical Cauchy number of the second kind, whose generating function is given by By using the corresponding lemma to Lemma 2, where ( ) is replaced bŷ( ) = Lif (− ln(1 + )), we can obtain the following result.

Theorem 7. For an integer and a nonnegative integer , one has
Putting = 1 in Theorem 7, one has the following.
Consider the case = 2. Note that the generating function of poly-Cauchy polynomial of the second kind ( ) ( ) [13] is given by Hence,̂( On the other hand, Thus,̂( Table 2). Theorem 9. For ≥ 0 and ≥ 1 one haŝ Putting = 1 in (80), we have the following identity. This is also an analogous formula to Euler's formula (14).

Theorem 13. For an integer and a nonnegative integer , one has
Putting = 1 in Theorem 13, we have the following.
Theorem 17. For ≥ 0 and ≥ 1 one has Define Then we obtain the following. (102)

Further Study
Kamano [14] mentioned that explicit formulae of ( ) ( ) for ≥ 4 seemed to be complicated to describe. We will give explicit formulae of ( ) ( ) for any ≥ 2 later anywhere else. In addition, one may consider the sums of products of ( − ) Cauchy numbers and poly-Cauchy numbers. It would be an interesting work to establish the explicit expressions of such summations.