1 Introduction

The Cauchy numbers (of the first kind) c n are given by the integral of the falling factorial:

$$c_n=\int_0^1 x(x-1) \cdots(x-n+1)\,dx=n!\int_0^1\binom{x}{n} \,dx $$

[7, Chap. VII]. The numbers c n /n! are sometimes called the Bernoulli numbers of the second kind (see, e.g., [2, 24]). Such numbers have been studied by several authors [6, 20, 21, 23, 25] because they are related to various special combinatorial numbers, including Stirling numbers of both kinds, Bernoulli numbers and harmonic numbers. Remarkably, the Cauchy numbers of the first kind c n and the Bernoulli numbers B n have several symmetric properties. The generating function of the Cauchy numbers of the first kind c n is given by

$$\frac{x}{\ln(1+x)}=\sum_{n=0}^\infty c_n\frac{x^n}{n!} $$

[7, 21], and the generating function of Bernoulli numbers B n is given by

$$\frac{x}{e^x-1}=\sum_{n=0}^\infty{B}_n\frac{x^n}{n!} $$

[7] or

$$\frac{x}{1-e^{-x}}=\sum_{n=0}^\infty B_n\frac{x^n}{n!} $$

[16]. In this paper we use the latter definition of B n . In addition, Cauchy numbers of the first kind c n can be written explicitly as

$$c_n=(-1)^n\sum_{m=0}^n \left[{n\atop m}\right]\frac{(-1)^m}{m+1} $$

([7, Chap. VII], [21, p. 1908]), where \(\bigl[{n\atop m} \bigr]\) are the (unsigned) Stirling numbers of the first kind, arising as coefficients of the rising factorial

$$x(x+1)\cdots(x+n-1)=\sum_{m=0}^n \left[{n\atop m}\right] x^m $$

(see, e.g., [9]). Bernoulli numbers B n (in the latter definition) can be also written explicitly as

$$B_n=(-1)^n\sum_{m=0}^n \left\{{n\atop m} \right\} \frac{(-1)^{m}m!}{m+1}, $$

where \(\bigl\{{n\atop m} \bigr\} \) are the Stirling numbers of the second kind, determined by

$$\left\{{n\atop m} \right\} =\frac{1}{m!}\sum_{j=0}^m(-1)^j \binom{m}{j}(m-j)^n $$

(see, e.g., [9]). Recently, Liu, Qi and Ding [20] established some recurrence relations about Cauchy numbers of the first kind as analogues of results for Bernoulli numbers by Agoh and Dilcher [1].

In 1997 Kaneko [16] introduced the poly-Bernoulli numbers \(B_{n}^{(k)}\) (n≥0, k≥1) by the generating function

$$\frac{{\rm Li}_k(1-e^{-x})}{1-e^{-x}}=\sum_{n=0}^\infty B_n^{(k)}\frac{x^n}{n!}, $$

where

$${\rm Li}_k(z)=\sum_{m=1}^\infty \frac{z^m}{m^k} $$

is the kth polylogarithm function. When k=1, \(B_{n}^{(1)}=B_{n}\) is the classical Bernoulli number with \(B_{1}^{(1)}=1/2\).

Recently, the author [17] introduced the poly-Cauchy numbers (of the first kind) \(c_{n}^{(k)}\) as a generalization of the Cauchy numbers and an analogue of the poly-Bernoulli numbers by

In addition, the generating function of poly-Cauchy numbers is given by

$${\rm Lif}_k\bigl(\ln(1+x)\bigr)=\sum_{n=0}^\infty c_n^{(k)}\frac{x^n}{n!}, $$

where

$${\rm Lif}_k(z):=\sum_{m=0}^\infty \frac{z^m}{m!(m+1)^k} $$

is the kth polylogarithm factorial function, which is also introduced by the author [17, 18]. If k=1, then \(c_{n}^{(1)}=c_{n}\) is the classical Cauchy number. One different extension of Cauchy numbers is on hypergeometric Cauchy numbers [19], as that of hypergeometric Bernoulli numbers is a different extension of Bernoulli numbers (e.g., [12, 13]).

The concept of the poly-Bernoulli numbers have been extended by several authors, including Bayad and Hamahata [3, 4], Hamahata and Masubuchi [10, 11], Sasaki [22] and Jolany [14]. Some applications of the poly-Bernoulli numbers have been studied (e.g., [5, 15]).

In this paper, we give a generalization of the poly-Cauchy numbers and show several combinatorial properties. The poly-Cauchy numbers are special ones with q=1 in the poly-Cauchy numbers with q parameter.

2 Poly-Cauchy numbers with q parameter

Let n and k be integers with n≥0 and k≥1. Let q be a real number with q≠0. Define the poly-Cauchy numbers with q parameter of the first kind \(c_{n,q}^{(k)}\) by

Hence, if q=1, then \(c_{n,1}^{(k)}=c_{n}^{(k)}\) are the poly-Cauchy numbers, defined in [17]. We may define the Cauchy numbers with q parameter of the first kind \(c_{n,q}^{(1)}=c_{n,q}\) by

$$c_{n,q}=\int_0^1 x(x-q)\cdots \bigl(x-(n-1)q\bigr)\,dx. $$

We record the first several Cauchy numbers with q parameter of the first kind:

As a general case of the poly-Cauchy numbers and the Cauchy numbers, the poly-Cauchy numbers with q parameter \(c_{n,q}^{(k)}\) can be expressed in terms of the (unsigned) Stirling numbers of the first kind \(\bigl[{n\atop m} \bigr]\).

Theorem 1

For a real number q≠0,

$$c_{n,q}^{(k)}=\sum_{m=0}^n \left[{n\atop m} \right]\frac {(-q)^{n-m}}{(m+1)^k}\quad(n\ge0,\ k\ge1). $$

Proof

By the identity

$$x(x-1)\cdots(x-n+1)=\sum_{m=0}^n \left[{n\atop m} \right](-1)^{n-m}x^m $$

(see, e.g., [9, Chap. 6]), we get

Hence, putting X=x 1 x 2⋯x k , we have

 □

We also obtain the generating function of the poly-Cauchy numbers with q parameter by using the polylogarithm factorial function \({\rm Lif}_{k}(z)\) [17, 18] defined by

$${\rm Lif}_k(z):=\sum_{m=0}^\infty \frac{z^m}{m!(m+1)^k}. $$

We may define the poly-Cauchy numbers with q parameter by the generating function. If q=1, the result is reduced to that of poly-Cauchy numbers.

Theorem 2

The generating function of the poly-Cauchy numbers with q parameter \(c_{n,q}^{(k)}\) is given by

$${\rm Lif}_k \biggl(\frac{\ln(1+qx)}{q} \biggr)=\sum _{n=0}^\infty c_{n,q}^{(k)} \frac{x^n}{n!}\quad(q\ne0). $$

Proof

Since

$$\frac{(\ln(1+x))^m}{m!}=(-1)^m\sum_{n=m}^\infty \left[{n\atop m} \right]\frac{(-x)^n}{n!}, $$

by Theorem 1 we have

 □

The generating function of the poly-Cauchy numbers with q parameter in Theorem 2 can be also written in the form of iterated integrals as that of the poly-Cauchy numbers.

Corollary 1

For k=1, we have

$$f_q(x):=\frac{q((1+q x)^{1/q}-1)}{\ln(1+q x)}=\sum_{n=0}^\infty c_{n,q}\frac{x^n}{n!}. $$

For k≥2, we have

Proof

Since

$${\rm Lif}_k(z)=\frac{1}{z}\int_0^z{ \rm Lif}_{k-1}(t)\,dt\quad(k\ge2) $$

with \({\rm Lif}_{1}(z)=(e^{z}-1)/z\), for k≥2, we have

$${\rm Lif}_k(z)=\underbrace{\frac{1}{z}\int _0^z\frac{1}{z}\int _0^z\cdots\frac{1}{z}\int _0^z}_{k-1}\frac{e^z-1}{z} \,\underbrace{dz \,dz\,\cdots \,dz}_{k-1}. $$

Putting z=ln(1+qx)/q, we get the result for k≥2.

For k=1, we have

$${\rm Lif}_1 \biggl(\frac{\ln(1+q x)}{q} \biggr)=\frac{q((1+q x)^{1/q}-1)}{\ln(1+q x)}. $$

 □

For q=1, we have

$$\sum_{m=0}^n \left\{{n\atop m} \right\} c_{m,1}^{(k)}=\frac{1}{(n+1)^k} $$

[17, Theorem 3]. However, we have not had a simple form of \(\sum_{m=0}^{n} \bigl\{{n\atop m} \bigr\} c_{m,q}^{(k)}\) for general q.

3 Poly-Cauchy numbers with q parameter of the second kind

The Cauchy numbers of the second kind \(\hat{c}_{n}\) are defined by

$$\hat{c}_n=\int_0^1(-x) (-x-1) \cdots(-x-n+1)\,dx $$

[7, Chap. VII]. Poly-Cauchy numbers of the second kind \(\hat{c}_{n}^{(k)}\) are defined by

[17]. Now, we define the poly-Cauchy numbers with q parameter of the second kind \(c_{n,q}^{(k)}\) by

Therefore, if q=1, then \(\hat{c}_{n,1}^{(k)}=\hat{c}_{n}^{(k)}\) are the poly-Cauchy numbers of the second kind. If q=k=1, then \(\hat{c}_{n,1}^{(1)}=\hat{c}_{n}\) are the Cauchy numbers of the second kind. In addition, we shall call \(\hat{c}_{n,q}^{(1)}=\hat{c}_{n,q}\) as the Cauchy numbers with q parameter of the second kind. We record the first several Cauchy numbers with q parameter of the second kind:

Similarly to the Poly-Cauchy numbers with q parameter of the first kind, the poly-Cauchy numbers with q parameter of the second kind can be expressed in terms of the Stirling numbers of the first kind. This is a general case of the results by Merlini et al. [21] for k=q=1 and by the author [17] for q=1. The proof is similar to that of Theorem 1 and omitted.

Theorem 3

For a real number q≠0,

$$\hat{c}_{n,q}^{(k)}=(-1)^n\sum _{m=0}^n \left[{n\atop m} \right]\frac {q^{n-m}}{(m+1)^k}\quad(n\ge0,\ k\ge1). $$

The generating function of the poly-Cauchy numbers with q parameter of the second kind \(\hat{c}_{n,q}^{(k)}\) can be also expressed by using the polylogarithm factorial function \({\rm Lif}_{k}(z)\). If q=1, then this generating function is reduced to that of poly-Cauchy numbers of the second kind \(\hat{c}_{n}^{(k)}=\hat{c}_{n,q}^{(k)}\).

Theorem 4

The generating function of the poly-Cauchy numbers with q parameter of the second kind \(\hat{c}_{n,q}^{(k)}\) is given by

$${\rm Lif}_k \biggl(-\frac{\ln(1+qx)}{q} \biggr)=\sum _{n=0}^\infty \hat{c}_{n,q}^{(k)} \frac{x^n}{n!}\quad(q\ne0). $$

The generating function of the poly-Cauchy numbers of the second kind can be also written in the form of iterated integrals by putting z=−ln(1+qx)/q in

$${\rm Lif}_k(z)=\underbrace{\frac{1}{z}\int _0^z\frac{1}{z}\int _0^z\cdots\frac{1}{z}\int _0^z}_{k-1}\frac{e^z-1}{z} \,\underbrace{dz \,dz\,\cdots\, dz}_{k-1}. $$

Corollary 2

For k=1, we have

$$g_q(x):=\frac{q(1-(1+q x)^{-1/q})}{\ln(1+q x)}=\sum_{n=0}^\infty \hat{c}_{n,q}\frac{x^n}{n!}. $$

For k≥2, we have

For q=1, we have

$$\sum_{m=0}^n \left\{{n\atop m} \right\} \hat{c}_{m,1}^{(k)}=\frac {(-1)^n}{(n+1)^k} $$

[17, Theorem 6]. However, we have not had a simple form of \(\sum_{m=0}^{n} \bigl\{{n\atop m} \bigr\} \hat{c}_{m,q}^{(k)}\) for general q.

4 Poly-Cauchy polynomials with q parameter

Define the poly-Cauchy polynomials with q parameter of the first kind \(c_{n,q}^{(k)}(z)\) and of the second kind \(\hat{c}_{n,q}^{(k)}(z)\) by

and

respectively. If q=1, then \(c_{n,1}^{(k)}(z)=c_{n}^{(k)}(z)\) and \(\hat{c}_{n,1}^{(k)}(z)=\hat{c}_{n}^{(k)}(z)\) are poly-Cauchy polynomials of the first kind and of the second kind, respectively [18]. Note that we also have a different definition where z and −z are interchanged [18]. If z=0, then \(c_{n,q}^{(k)}(0)=c_{n,q}^{(k)}\) and \(\hat{c}_{n,q}^{(k)}(0)=\hat{c}_{n,q}^{(k)}\) are poly-Cauchy numbers with q parameter of the first kind and of the second kind, respectively.

Poly-Cauchy polynomials with q parameter of the first kind \(c_{n,q}^{(k)}(z)\) and of the second kind \(\hat{c}_{n,q}^{(k)}(z)\) are expressed by using the (unsigned) Stirling numbers of the first kind \(\bigl[{n\atop m} \bigr]\).

Theorem 5

For integers n and k with n≥0 and k≥1 and a real number q≠0, we have

Proof

Similarly to the proof of Theorem 1, putting X=x 1 x 2⋯x k −z, we have

The second identity is proven similarly and omitted. □

Theorem 6

Let n and k be integers with n≥0 and k≥1, and q be a real number with q≠0. Then the generating functions of the poly-Cauchy polynomials with q parameter of the first kind \(c_{n,q}^{(k)}(z)\) and of the second kind \(\hat{c}_{n,q}^{(k)}(z)\) are given by

$$(1+q x)^{-z/q}{\rm Lif}_k \biggl(\frac{\ln(1+q x)}{q} \biggr)= \sum_{n=0}^\infty c_{n,q}^{(k)}(z) \frac{x^n}{n!} $$

and

$$(1+q x)^{z/q}{\rm Lif}_k \biggl(-\frac{\ln(1+q x)}{q} \biggr)=\sum_{n=0}^\infty\hat{c}_{n,q}^{(k)}(z)\frac{x^n}{n!}, $$

respectively.

Proof

Similarly to the proof of Theorem 2, by the first identity of Theorem 5 we have

The second identity is proven similarly and omitted. □

Therefore, by Corollaries 1 and 2 with Theorem 6 we obtain the generating function of poly-Cauchy polynomials with q parameter in the form of iterated integrals. Let f q (x) and g q (x) be as in Corollaries 1 and 2, respectively.

Corollary 3

For k=1, we have

For k≥2, we have

5 Some properties of poly-Cauchy numbers and polynomials with q parameter

It is known that poly-Bernoulli numbers satisfy the duality theorem \(B_{n}^{(-k)}=B_{k}^{(-n)}\) for n,k≥0 [16, Theorem 2] because of the symmetric formula

$$\sum_{n=0}^\infty\sum _{k=0}^\infty B_n^{(-k)} \frac{x^n}{n!}\frac {y^k}{k!}=\frac{e^{x+y}}{e^x+e^y-e^{x+y}}. $$

However, poly-Cauchy numbers with q parameter do not satisfy the duality theorem for any q≠0, by the following results.

Proposition 1

For nonnegative integers n and k and a real number q≠0, we have

Proof

We shall prove the first identity. The second identity is proven similarly. By Theorem 2 we have

 □

Poly-Bernoulli polynomials \(B_{n}^{(k)}(z)\) are defined as

$$\frac{{\rm Li}_k(1-e^{-x})}{1-e^{-x}}e^{xz}=\sum_{n=0}^\infty B_n^{(k)}(z)\frac{x^n}{n!} $$

[3]. Note that \(B_{n}^{(k)}(z)\) are defined in [8] by replacing e xz by e −xz. Concerning the poly-Bernoulli polynomials, for an integer k and a positive integer n, we have

$$\frac{d}{dz}B_n^{(k)}(z)=n B_{n-1}^{(k)}(z) $$

[3, Theorem 1.4]. However, poly-Cauchy polynomials with q parameter are not such sequences for any q≠0. By differentiating \(c_{n,q}^{(k)}(z)\) or \(\hat{c}_{n,q}^{(k)}(z)\), we have the following:

Proposition 2

For nonnegative integers n and k and a real number q≠0, we have

Proof

Differentiating both sides of the first identity in Theorem 6 with respect to z, we have

$$-\frac{\ln(1+q x)}{q}(1+q x)^{-z/q}{\rm Lif}_k \biggl( \frac{\ln(1+q x)}{q} \biggr)=\sum_{n=0}^\infty \frac{d}{dz}c_{n,q}^{(k)}(z)\frac{x^n}{n!}. $$

Then,

and

$${\rm RHS}=\sum_{n=1}^\infty \frac{d}{dz}c_{n,q}^{(k)}(z)\frac{x^n}{n!}. $$

The second identity is proven similarly and omitted. □

Lastly, we show a recurrence formula for the poly-Cauchy polynomials \(c_{n}^{(k)}(z)=c_{n,1}^{(k)}(z)\) in terms of the poly-Cauchy numbers \(c_{n}^{(k)}=c_{n,1}^{(k)}\) and the Cauchy polynomials \(c_{n}(z)=c_{n,1}^{(1)}(z)\).

Theorem 7

For integers n and k with n≥0 and k≥1, we have

Proof

$$\frac{d}{dz}\bigl(z{\rm Lif}_k(z)\bigr)={\rm Lif}_{k-1}(z), $$

so

$$ {\rm Lif}_k(z)=\frac{1}{z}\int_0^z{\rm Lif}_{k-1}(t)\,dt. $$
(1)

If we put z=ln(1+x) and t=log(1+s) in the identity, then

$$\frac{{\rm Lif}_k(\ln(1+x))}{(1+x)^z}=\frac{1}{(1+x)^z\ln(1+x)}\int_0^x \frac{{\rm Lif}_{k-1}(\ln(1+s))}{1+s}\,ds. $$

By the generating function in Theorem (6),

The second recurrence relation is obtained similarly. □

6 Some more extensions

We shall consider integrals of the definition of poly-Cauchy numbers with q parameter in the range [0,l], where l is a real number with l≠0 instead of the range [0,1]. Define \(c_{n,q}^{(k)}(l_{1},l_{2},\ldots,l_{k})\), where l 1,l 2,…,l k are nonzero real numbers, by

Then \(c_{n,q}^{(k)}(l_{1},l_{2},\dots,l_{k})\) can be also expressed in terms of the (unsigned) Stirling numbers of the first kind \(\bigl[{n\atop m} \bigr]\).

Theorem 8

For a real number q≠0,

$$c_{n,q}^{(k)}(l_1,l_2, \dots,l_k)=\sum_{m=0}^n \left[{n\atop m} \right]\frac{(-q)^{n-m}(l_1 l_2\cdots l_k)^{m+1}}{(m+1)^k}\quad(n\ge0,\, k\ge1). $$

The numbers l 1,l 2,…,l k and q are not necessarily positive integers. For example, for k=2, n=4, \(l_{1}=\sqrt{2}\) and l 2=−1/3, we have

If k=q=1 in Theorem 8, then

$$c_{n,1}^{(1)}(l)=\sum_{m=0}^n \left[{n\atop m} \right]\frac {(-1)^{n-m}l^{m+1}}{m+1}, $$

which is the relation in [21, Corollary 3.3].

The generating function of \(c_{n,q}^{(k)}(l_{1},l_{2},\dots,l_{k})\) is also given by using the polylogarithm factorial function \({\rm Lif}_{k}(z)\).

Theorem 9

For a real number q≠0,

$$l_1 l_2\cdots l_k\cdot{\rm Lif}_k \biggl(\frac{l_1 l_2\dots l_k\ln (1+qx)}{q} \biggr)=\sum_{n=0}^\infty c_{n,q}^{(k)}(l_1,l_2,\dots ,l_k)\frac{x^n}{n!}\quad(n\ge0,\,k\ge1). $$

If k=q=1 in Theorem 9, then

which is the relation in [21, Theorem 3.2].

The generating function of \(c_{n,q}^{(k)}(l_{1},l_{2},\dots,l_{k})\) in Theorem 9 can be also written in the form of iterated integrals.

Corollary 4

For k=1, we have

$$f_q(l_1,\ldots,l_k;x):= \frac{q((1+q x)^{l_1 l_2\cdots l_k/q}-1)}{\ln(1+q x)}= \sum_{n=0}^\infty c_{n,q}^{(1)}(l_1,l_2, \ldots,l_k)\frac{x^n}{n!}. $$

For k≥2, we have

In similar manners, if we define \(\hat{c}_{n,q}^{(k)}(l_{1},l_{2},\dots ,l_{k})\) by

then we have the following series of results.

Theorem 10

For a real number q≠0,

$$\hat{c}_{n,q}^{(k)}(l_1,l_2, \dots,l_k)=(-1)^n\sum_{m=0}^n \left[{n\atop m}\right ]\frac{q^{n-m}(l_1 l_2\cdots l_k)^{m+1}}{(m+1)^k}\quad(n\ge 0,\,k\ge1). $$

Theorem 11

The generating function of \(\hat{c}_{n,q}^{(k)}(l_{1},l_{2},\ldots,l_{k})\) (q≠0) is given by

$$l_1 l_2\cdots l_k\cdot{\rm Lif}_k \biggl(-l_1 l_2\cdots l_k\frac{\ln (1+qx)}{q} \biggr)=\sum_{n=0}^\infty\hat{c}_{n,q}^{(k)}(l_1,l_2,\ldots ,l_k)\frac{x^n}{n!}\quad(n\ge0,\,k\ge1). $$

Corollary 5

For k=1, we have

$$g_q(l_1,\ldots,l_k;x):= \frac{q(1-(1+q x)^{-l_1 l_2\cdots l_k/q})}{\ln(1+q x)}= \sum_{n=0}^\infty\hat{c}_{n,q}^{(1)}(l_1,l_2, \ldots,l_k)\frac{x^n}{n!}. $$

For k≥2, we have

Polynomials \(c_{n,q}^{(k)}(z;l_{1},l_{2},\ldots,l_{k})\) and \(\hat{c}_{n,q}^{(k)}(z;l_{1},l_{2},\ldots,l_{k})\) are similarly defined, and their explicit formulae and generating functions are obtained by replacing the range of the integral [0,1] by [0,l].

7 Future work

There is the following relation between poly-Cauchy numbers and poly-Bernoulli numbers [17, Theorem 8]:

$$B_n^{(k)}=\sum_{l=1}^n \sum_{m=1}^n m! \left\{{n\atop m} \right\} \left\{{m-1\atop l-1} \right\} c_l^{(k)}\quad(n\ge1). $$

However, any corresponding generalized poly-Bernoulli numbers to poly-Cauchy numbers with q parameter have not yet been studied, though one candidate may be

$$B_{n,q}^{(k)}=\sum_{m=0}^n \left\{{n\atop m} \right\} \frac {(-q)^{n-m}m!}{(m+1)^k}. $$

On the other hand, though several generalized poly-Bernoulli numbers have been studied (see, e.g., [3, 4, 10, 11, 14, 22, 23]), the corresponding generalized poly-Cauchy numbers have not yet been studied, either. One of the reasons is that the method to generalize the poly-Cauchy numbers in this paper is based upon the definition of integrals and the methods to generalize the poly-Bernoulli numbers in other works are based upon the definition of generating functions.