Abstract

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 analogous results by investigating a general type of sums of products of Cauchy numbers including poly-Cauchy numbers: 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.

2. 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: where Bernoulli numbers are defined by the generating function (3) and poly-Bernoulli numbers are defined by the generating function (9) and 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:

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

Put

Since we have

Since the coefficient of in 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 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.

Now, by the generating function (21), the identity (25), and Lemma 2,

Note that () and (). Therefore,

If we put in Theorem 1, we get an analogous formula to Euler's formula (14) for sums of products of Cauchy number and a poly-Cauchy number.

Corollary 3. One has

2.1. Explicit Formula for

Theorem 1 gives only relations among sums of products . For , an explicit formula for is given.

Theorem 4. For and one has where .

Proof. Consider 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 we have
Hence,
Therefore, we get the identity (40).
Finally, by we have
Hence,
Therefore, we get the identity (41).

Putting 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)

Table 1 

Proof. Since we have . Hence,

2.2. Explicit Formulae for

For , an explicit formula for is also given.

Theorem 6. For and one has

Proof. Consider
Hence, by Corollary 5 which is the identity (55).
By Lemma 2 with
Thus,
By multiplying both sides by and summing over from to , we obtain
Hence, we have
By comparing the coefficients of in both sides,
Dividing both sides by , we have the identity (56).
Finally, since we obtain
Hence, we have yielding the identity (57).

2.3. Poly-Cauchy Numbers of the Second Kind

Similarly, define by where is poly-Cauchy number of the second kind [12], whose generating function is given by can be also defined by (see [12]). In this sense, is called poly-Cauchy number of the first kind. When , 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 by , we can obtain the following result.

Theorem 7. For an integer and a nonnegative integer , one has

Putting in Theorem 7, one has the following.

Corollary 8.

Consider the case . Note that the generating function of poly-Cauchy polynomial of the second kind [13] is given by are expressed explicitly in terms of the Stirling numbers of the first kind [13, Theorem 4]

Hence,

On the other hand,

Thus, (see Table 2).

Table 2 

Theorem 9. For and one has

Putting in (80), we have the following identity. This is also an analogous formula to Euler’s formula (14).

Corollary 10. One has

If , then can be expressed as follows.

Theorem 11. For and one has

Remark 12. Note that

2.4. Two Kinds of Poly-Cauchy Numbers

Define by

Then we obtain the following.

Theorem 13. For an integer and a nonnegative integer , one has

Putting in Theorem 13, we have the following.

Corollary 14.

If , then can be expressed explicitly.

Theorem 15. For and one has

Putting in (89), we have the alternative identity (2.3) in [6, Theorem 2.4] because by (2.2) in [6, Theorem 2.4].

Corollary 16. One has

If , then can be expressed as follows.

Theorem 17. For and one has

Define by

Then we obtain the following.

Theorem 18. For an integer and a nonnegative integer , one has

Putting in Theorem 18, we have the following.

Corollary 19.

If , then can be expressed explicitly.

Theorem 20. For and one has

Putting in (99), we have the identity (2.3) in [6, Theorem 2.4].

Corollary 21. One has

If , then can be expressed as follows.

Theorem 22. For and one has

3. Further Study

Kamano [14] mentioned that explicit formulae of for seemed to be complicated to describe. We will give explicit formulae of for any 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.

Acknowledgments

This work was supported in part by the Grant-in-Aid for Scientific research (C) (no. 22540005), the Japan Society for the Promotion of Science.