A note on poly-Bernoulli numbers and polynomials of the second kind

Kim, Taekyun; Kwon, Hyuck In; Lee, Sang Hun; Seo, Jong Jin

doi:10.1186/1687-1847-2014-219

A note on poly-Bernoulli numbers and polynomials of the second kind

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Published: 05 August 2014

Volume 2014, article number 219, (2014)
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A note on poly-Bernoulli numbers and polynomials of the second kind

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Taekyun Kim¹,
Hyuck In Kwon¹,
Sang Hun Lee² &
…
Jong Jin Seo³

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Abstract

In this paper, we consider the poly-Bernoulli numbers and polynomials of the second kind and presents new and explicit formulas for calculating the poly-Bernoulli numbers of the second kind and the Stirling numbers of the second kind.

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1 Introduction

As is well known, the Bernoulli polynomials of the second kind are defined by the generating function to be

\frac{t}{log (1 + t)} {(1 + t)}^{x} = \sum_{n = 0}^{\infty} b_{n} (x) \frac{t^{n}}{n!} (see [1–3]) .

(1)

When $x = 0$ , $b_{n} = b_{n} (0)$ are called the Bernoulli numbers of the second kind. The first few Bernoulli numbers $b_{n}$ of the second kind are $b_{0} = 1$ , $b_{1} = 1 / 2$ , $b_{2} = - 1 / 12$ , $b_{3} = 1 / 24$ , $b_{4} = - 19 / 720$ , $b_{5} = 3 / 160, \dots$ .

From (1), we have

b_{n} (x) = \sum_{l = 0}^{n} (\binom{n}{l}) b_{l} {(x)}_{n - l},

(2)

where ${(x)}_{n} = x (x - 1) \dots (x - n + 1)$ ( $n ≧ 0$ ). The Stirling number of the second kind is defined by

x^{n} = \sum_{l = 0}^{n} S_{2} (n, l) {(x)}_{l} (n ≧ 0) .

(3)

The ordinary Bernoulli polynomials are given by

\frac{t}{e^{t} - 1} e^{x t} = \sum_{n = 0}^{\infty} B_{n} (x) \frac{t^{n}}{n!} (see [1–18]) .

(4)

When $x = 0$ , $B_{n} = B_{n} (0)$ are called Bernoulli numbers.

It is well known that the classical poly-logarithmic function ${Li}_{k} (x)$ is given by

{Li}_{k} (x) = \sum_{n = 1}^{\infty} \frac{x^{n}}{n^{k}} (k \in Z) (see [8–10]) .

(5)

For $k = 1$ , ${Li}_{1} (x) = \sum_{n = 1}^{\infty} \frac{x^{n}}{n} = - log (1 - x)$ . The Stirling number of the first kind is defined by

{(x)}_{n} = \sum_{l = 0}^{n} S_{1} (n, l) x^{l} (n \geq 0) (see [16]) .

(6)

In this paper, we consider the poly-Bernoulli numbers and polynomials of the second kind and presents new and explicit formulas for calculating the poly-Bernoulli number and polynomial and the Stirling number of the second kind.

2 Poly-Bernoulli numbers and polynomials of the second kind

For $k \in Z$ , we consider the poly-Bernoulli polynomials $b_{n}^{(k)} (x)$ of the second kind:

\frac{{Li}_{k} (1 - e^{- t})}{log (1 + t)} {(1 + t)}^{x} = \sum_{n = 0}^{\infty} b_{n}^{(k)} (x) \frac{t^{n}}{n!} .

(7)

When $x = 0$ , $b_{n}^{(k)} = b_{n}^{(k)} (0)$ are called the poly-Bernoulli numbers of the second kind.

Indeed, for $k = 1$ , we have

\frac{{Li}_{k} (1 - e^{- t})}{log (1 + t)} {(1 + t)}^{x} = \frac{t}{log (1 + t)} {(1 + t)}^{x} = \sum_{n = 0}^{\infty} b_{n} (x) \frac{t^{n}}{n!} .

(8)

By (7) and (8), we get

b_{n}^{(1)} (x) = b_{n} (x) (n \geq 0) .

(9)

It is well known that

\frac{t {(1 + t)}^{x - 1}}{log (1 + t)} = \sum_{n = 0}^{\infty} B_{n}^{(n)} (x) \frac{t^{n}}{n!},

(10)

where $B_{n}^{(α)} (x)$ are the Bernoulli polynomials of order α which are given by the generating function to be

{(\frac{t}{e^{t} - 1})}^{α} e^{x t} = \sum_{n = 0}^{\infty} B_{n}^{(α)} (x) \frac{t^{n}}{n!} (see [1–18]) .

By (1) and (10), we get

b_{n} (x) = B_{n}^{(n)} (x + 1) (n \geq 0) .

Now, we observe that

(11)

Thus, by (11), we get

\begin{aligned} \sum_{n = 0}^{\infty} b_{n}^{(2)} (x) \frac{t^{n}}{n!} & = \frac{{(1 + t)}^{x}}{log (1 + t)} \int_{0}^{t} \frac{x}{e^{x} - 1} d x \\ = \frac{{(1 + t)}^{x}}{log (1 + t)} \sum_{l = 0}^{\infty} \frac{B_{l}}{l!} \int_{0}^{t} x^{l} d x \\ = (\frac{t}{log (1 + t)}) {(1 + t)}^{x} \sum_{l = 0}^{\infty} \frac{B_{l}}{(l + 1)} \frac{t^{l}}{l!} \\ = \sum_{n = 0}^{\infty} {\sum_{l = 0}^{n} (\binom{n}{l}) \frac{B_{l} b_{n - l} (x)}{l + 1}} \frac{t^{n}}{n!} . \end{aligned}

(12)

Therefore, by (12), we obtain the following theorem.

Theorem 2.1 For $n \geq 0$ we have

b_{n}^{(2)} (x) = \sum_{l = 0}^{n} (\binom{n}{l}) \frac{B_{l} b_{n - l} (x)}{l + 1} .

From (11), we have

\begin{aligned} \sum_{n = 0}^{\infty} b_{n}^{(k)} (x) \frac{t^{n}}{n!} & = \frac{{Li}_{k} (1 - e^{- t})}{log (1 + t)} {(1 + t)}^{x} \\ = \frac{t}{log (1 + t)} \frac{{Li}_{k} (1 - e^{- t})}{t} {(1 + t)}^{x} . \end{aligned}

(13)

We observe that

\begin{aligned} \frac{1}{t} {Li}_{k} (1 - e^{- t}) & = \frac{1}{t} \sum_{n = 1}^{\infty} \frac{1}{n^{k}} {(1 - e^{- t})}^{n} \\ = \frac{1}{t} \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n}}{n^{k}} n! \sum_{l = n}^{\infty} S_{2} (l, n) \frac{{(- t)}^{l}}{l!} \\ = \frac{1}{t} \sum_{l = 1}^{\infty} \sum_{n = 1}^{l} \frac{{(- 1)}^{n + l}}{n^{k}} n! S_{2} (l, n) \frac{t^{l}}{l!} \\ = \sum_{l = 0}^{\infty} \sum_{n = 1}^{l + 1} \frac{{(- 1)}^{n + l + 1}}{n^{k}} n! \frac{S_{2} (l + 1, n)}{l + 1} \frac{t^{l}}{l!} . \end{aligned}

(14)

Thus, by (10) and (14), we get

\begin{aligned} \sum_{n = 0}^{\infty} b_{n}^{(k)} (x) \frac{t^{n}}{n!} & = (\sum_{m = 0}^{\infty} b_{m} (x) \frac{t^{m}}{m!}) {\sum_{l = 0}^{\infty} (\sum_{p = 1}^{l + 1} \frac{{(- 1)}^{p + l + 1}}{p^{k}} p! \frac{S_{2} (l + 1, p)}{l + 1}) \frac{t^{l}}{l!}} \\ = \sum_{n = 0}^{\infty} {\sum_{l = 0}^{n} (\binom{n}{l}) (\sum_{p = 1}^{l + 1} \frac{{(- 1)}^{p + l + 1} p!}{p^{k}} \frac{S_{2} (l + 1, p)}{l + 1}) b_{n - l} (x)} \frac{t^{n}}{n!} . \end{aligned}

(15)

Therefore, by (15), we obtain the following theorem.

Theorem 2.2 For $n \geq 0$ , we have

b_{n}^{(k)} (x) = \sum_{l = 0}^{n} (\binom{n}{l}) (\sum_{p = 1}^{l + 1} \frac{{(- 1)}^{p + l + 1}}{p^{k}} p! \frac{S_{2} (l + 1, p)}{l + 1}) b_{n - l} (x) .

By (7), we get

\begin{aligned} \sum_{n = 0}^{\infty} (b_{n}^{(k)} (x + 1) - b_{n}^{(k)} (x)) \frac{t^{n}}{n!} & = \frac{{Li}_{k} (1 - e^{- t})}{log (1 + t)} {(1 + t)}^{x + 1} - \frac{{Li}_{k} (1 - e^{- t})}{log (1 + t)} {(1 + t)}^{x} \\ = \frac{t {Li}_{k} (1 - e^{- t})}{log (1 + t)} {(1 + t)}^{x} \\ = (\frac{t}{log (1 + t)} {(1 + t)}^{x}) {Li}_{k} (1 - e^{- t}) \\ = (\sum_{l = 0}^{\infty} \frac{b_{l} (x)}{l!} t^{l}) {\sum_{p = 1}^{\infty} (\sum_{m = 1}^{p} \frac{{(- 1)}^{m + p} m!}{m^{k}} S_{2} (p, m))} \frac{t^{p}}{p!} \end{aligned}

(16)

\begin{aligned} = \sum_{n = 1}^{\infty} (\sum_{p = 1}^{n} \sum_{m = 1}^{p} \frac{{(- 1)}^{m + p}}{m^{k}} m! S_{2} (p, m) \frac{b_{n - p} (x) n!}{(n - p)! p!}) \frac{t^{n}}{n!} \\ = \sum_{n = 1}^{\infty} {\sum_{p = 1}^{n} \sum_{m = 1}^{p} (\binom{n}{p}) \frac{{(- 1)}^{m + p} m!}{m^{k}} S_{2} (p, m) b_{n - p} (x)} \frac{t^{n}}{n!} . \end{aligned}

(17)

Therefore, by (16), we obtain the following theorem.

Theorem 2.3 For $n \geq 1$ , we have

b_{n}^{(k)} (x + 1) - b_{n}^{(k)} (x) = \sum_{p = 1}^{n} \sum_{m = 1}^{p} (\binom{n}{p}) \frac{{(- 1)}^{m + p} m!}{m^{k}} S_{2} (p, m) b_{n - p} (x) .

(18)

From (13), we have

\begin{aligned} \sum_{n = 0}^{\infty} b_{n}^{(k)} (x + y) \frac{t^{n}}{n!} & = {(\frac{{Li}_{k} (1 - e^{- t})}{log (1 + t)})}^{k} {(1 + t)}^{x + y} \\ = {(\frac{{Li}_{k} (1 - e^{- t})}{log (1 + t)})}^{k} {(1 + t)}^{x} {(1 + t)}^{y} \\ = (\sum_{l = 0}^{\infty} b_{l}^{(k)} (x) \frac{t^{l}}{l!}) (\sum_{m = 0}^{\infty} {(y)}_{m} \frac{t^{m}}{m!}) \\ = \sum_{n = 0}^{\infty} (\sum_{l = 0}^{n} {(y)}_{l} b_{n - l}^{(k)} (x) \frac{n!}{(n - l)! l!}) \frac{t^{n}}{n!} \\ = \sum_{n = 0}^{\infty} (\sum_{l = 0}^{n} (\binom{n}{l}) b_{n - l}^{(k)} (x) {(y)}_{l}) \frac{t^{n}}{n!} . \end{aligned}

(19)

Therefore, by (17), we obtain the following theorem.

Theorem 2.4 For $n \geq 0$ , we have

b_{n}^{(k)} (x + y) = \sum_{l = 0}^{n} (\binom{n}{l}) b_{n - l}^{(k)} (x) {(y)}_{l} .

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Acknowledgements

The present research has been conducted by the Research Grant of Kwangwoon University in 2014.

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Department of Mathematics, Kwangwoon University, Seoul, 139-701, Republic of Korea
Taekyun Kim & Hyuck In Kwon
Division of General Education, Kwangwoon University, Seoul, 139-701, Republic of Korea
Sang Hun Lee
Department of Applied Mathematics, Pukyong National University, Pusan, 698-737, Republic of Korea
Jong Jin Seo

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Taekyun Kim
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Hyuck In Kwon
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Kim, T., Kwon, H.I., Lee, S.H. et al. A note on poly-Bernoulli numbers and polynomials of the second kind. Adv Differ Equ 2014, 219 (2014). https://doi.org/10.1186/1687-1847-2014-219

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Received: 23 June 2014
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Published: 05 August 2014
DOI: https://doi.org/10.1186/1687-1847-2014-219

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A note on poly-Bernoulli numbers and polynomials of the second kind

Abstract

Similar content being viewed by others

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References

Acknowledgements

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