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A note on degenerate poly-Bernoulli numbers and polynomials

Advances in Difference Equations20152015:258

https://doi.org/10.1186/s13662-015-0595-3

Received: 18 March 2015

Accepted: 6 August 2015

Published: 20 August 2015

Abstract

In this paper, we consider the degenerate poly-Bernoulli polynomials and present new and explicit formulas for computing them in terms of the degenerate Bernoulli polynomials and Stirling numbers of the second kind.

Keywords

degenerate poly-Bernoulli polynomialdegenerate Bernoulli polynomialStirling number of the second kind

MSC

11B6811B7311B83

1 Introduction

For \(\lambda\in\mathbb{C}\), Carlitz considered the degenerate Bernoulli polynomials given by the generating function
$$ \frac{t}{ (1+\lambda t )^{\frac{1}{\lambda}}-1} (1+\lambda t )^{\frac{x}{\lambda}}=\sum _{n=0}^{\infty}\beta_{n} (x\mid \lambda ) \frac{t^{n}}{n!}\quad (\text{see [1--3]}). $$
(1.1)

When \(x=0\), \(\beta_{n} (\lambda )=\beta_{n} (0\mid\lambda )\) are called the degenerate Bernoulli numbers.

Thus, by (1.1), we get
$$ \beta_{n} (x\mid\lambda )=\sum_{l=0}^{n} \binom{n}{l}\beta _{l} (\lambda ) (x\mid\lambda )_{n-l}, $$
(1.2)
where \((x\mid\lambda )_{n}=x (x-\lambda ) (x-2\lambda )\cdots (x-\lambda (n-1 ) )\).
The classical polylogarithm function \(\operatorname{Li}_{k}\) is
$$ \operatorname{Li}_{k} (x )=\sum_{n=1}^{\infty} \frac{x^{n}}{n^{k}}\quad (k\in\mathbb{Z} \text{; see [2, 4--11]}). $$
(1.3)
From (1.1), we note that
$$\begin{aligned} & \sum_{n=0}^{\infty}\lim_{\lambda\rightarrow0} \beta _{n} (x\mid\lambda )\frac{t^{n}}{n!} \\ &\quad =\lim_{\lambda\rightarrow0}\frac{t}{ (1+\lambda t )^{\frac {1}{\lambda}}-1} (1+\lambda t )^{\frac{x}{\lambda}} \\ & \quad =\frac{t}{e^{t}-1}e^{xt} \\ & \quad =\sum_{n=0}^{\infty}B_{n} (x ) \frac{t^{n}}{n!}, \end{aligned}$$
(1.4)
where \(B_{n} (x )\) are called the Bernoulli polynomials (see [127]).
Thus, by (1.4), we get
$$ \lim_{\lambda\rightarrow0}\beta_{n} (x\mid\lambda )=B_{n} (x )\quad (n\ge0 ). $$
(1.5)
In [4, 14], the poly-Bernoulli polynomials are given by
$$ \frac{\operatorname{Li}_{k} (1-e^{-t} )}{e^{t}-1}e^{xt}=\sum_{n=0}^{\infty }B_{n}^{ (k )} (x )\frac{t^{n}}{n!}. $$
(1.6)
For \(k=1\), we have
$$\begin{aligned} \frac{\operatorname{Li}_{1} (1-e^{-t} )}{e^{t}-1}e^{xt}&=\frac {t}{e^{t}-1}e^{xt} \\ &=\sum _{n=0}^{\infty}B_{n} (x ) \frac {t^{n}}{n!}. \end{aligned}$$
(1.7)

By (1.4) and (1.7), we get \(B_{n}^{ (1 )} (x )=B_{n} (x )\).

The Stirling numbers of the second kind are given by
$$ x^{n}=\sum_{l=0}^{n}S_{2} (n,l ) (x )_{l} \quad (\text{see [1--27]}), $$
(1.8)
and the Stirling numbers of the first kind are defined by
$$ (x )_{n}=x (x-1 )\cdots (x-n+1 )=\sum_{l=0}^{n}S_{1} (n,l )x^{l}\quad (n\ge0 ). $$
(1.9)

The purpose of this paper is to construct the degenerate poly-Bernoulli polynomials and present new and explicit formulas for computing them in terms of the degenerate Bernoulli polynomials and Stirling numbers of the second kind.

2 Degenerate poly-Bernoulli numbers and polynomials

For \(\lambda\in\mathbb{C}\), \(k\in\mathbb{Z}\), we consider the degenerate poly-Bernoulli polynomials given by the generating function
$$ \frac{\operatorname{Li}_{k} (1-e^{-t} )}{ (1+\lambda t )^{\frac {1}{\lambda}}-1} (1+\lambda t )^{\frac{x}{\lambda}}=\sum _{n=0}^{\infty}\beta_{n}^{ (k )} (x\mid \lambda )\frac {t^{n}}{n!}. $$
(2.1)

When \(x=0\), \(\beta_{n}^{ (k )} (\lambda )=\beta _{n}^{ (k )} (0\mid\lambda )\) are called the degenerate poly-Bernoulli numbers. Note that \(\beta _{n}^{ (1 )} (x\mid\lambda )=\beta_{n} (x\mid \lambda )\) and \(\lim_{\lambda\rightarrow0}\beta_{n}^{ (k )} (x\mid \lambda )=B_{n}^{ (k )} (x )\).

From (2.1), we can derive the following equation:
$$\begin{aligned} \sum_{n=0}^{\infty}\beta_{n}^{ (k )} (x\mid\lambda )\frac{t^{n}}{n!} & = \biggl(\frac{\operatorname{Li}_{k} (1-e^{-t} )}{ (1+\lambda t )^{\frac{1}{\lambda}}-1} \biggr) (1+\lambda t )^{\frac{x}{\lambda}} \\ & = \Biggl(\sum_{l=0}^{\infty} \beta_{l}^{ (k )} (\lambda )\frac{t^{l}}{l!} \Biggr) \Biggl( \sum_{m=0}^{\infty} (x\mid \lambda )_{m}\frac{t^{m}}{m!} \Biggr) \\ & =\sum_{n=0}^{\infty} \Biggl(\sum _{l=0}^{n}\binom{n}{l}\beta_{l}^{ (k )} (\lambda ) (x\mid\lambda )_{n-l} \Biggr)\frac{t^{n}}{n!}. \end{aligned}$$
(2.2)
Thus, by (2.2), we get
$$ \beta_{n}^{ (k )} (x\mid\lambda )=\sum _{l=0}^{n}\binom {n}{l}\beta_{l}^{ (k )} (\lambda ) (x\mid\lambda )_{n-l}. $$
(2.3)
Now, we observe that
$$\begin{aligned} & \frac{\operatorname{Li}_{k} (1-e^{-t} )}{ (1+\lambda t )^{\frac{1}{\lambda}}-1} (1+t )^{\frac{x}{\lambda }} \\ &\quad =\sum_{n=0}^{\infty}\beta_{n}^{ (k )} (x\mid\lambda )\frac{t^{n}}{n!} \\ &\quad =\frac{ (1+t )^{\frac{x}{\lambda}}}{ (1+\lambda t )^{\frac{1}{\lambda}}-1}\underset{ (k-2)\text{ times}}{\int_{0}^{t} \underbrace{\frac{1}{e^{y}-1}\int_{0}^{y} \frac{1}{e^{y}-1}\int_{0}^{y}\cdots \frac{1}{e^{y}-1}\int_{0}^{y}}} \frac{y}{e^{y}-1}\,dy\cdots dy. \end{aligned}$$
(2.4)
From (2.4), we have
$$\begin{aligned} & \sum_{n=0}^{\infty}\beta_{n}^{ (2 )} (x\mid\lambda )\frac{t^{n}}{n!} \\ & \quad =\frac{ (1+t )^{\frac{x}{\lambda}}}{ (1+\lambda t )^{\frac{1}{\lambda}}-1}\int_{0}^{t} \frac{y}{e^{y}-1}\,dy \\ &\quad =\frac{ (1+t )^{\frac{x}{\lambda}}}{ (1+\lambda t )^{\frac{1}{\lambda}}-1}\sum_{l=0}^{\infty} \frac{B_{l}}{l!}\int_{0}^{t}y^{l}\,dy \\ &\quad = \biggl(\frac{t}{ (1+\lambda t )^{\frac{1}{\lambda}}-1} (1+\lambda t )^{\frac{x}{\lambda}} \biggr) \Biggl(\sum _{l=0}^{\infty }\frac{B_{l}}{l+1} \frac{t^{l}}{l!} \Biggr) \\ & \quad =\sum_{n=0}^{\infty} \Biggl\{ \sum _{l=0}^{n}\binom{n}{l}\frac {B_{l}}{l+1} \beta_{n-l} (x\mid\lambda ) \Biggr\} \frac {t^{n}}{n!}, \end{aligned}$$
(2.5)
where \(B_{n}=B_{n} (0 )\) are the Bernoulli numbers.

By comparing the coefficients on both sides of (2.5), we obtain the following theorem.

Theorem 2.1

For \(n\ge0\), we have
$$\begin{aligned} \begin{aligned} \beta_{n}^{ (2 )} (x\mid\lambda ) & =\sum _{l=0}^{n}\binom{n}{l}\frac{B_{l}}{l+1} \beta_{n-l} (x\mid\lambda ) \\ & =\beta_{n} (x\mid\lambda )-\frac{n}{4}\beta_{n-1} (x \mid \lambda )+\sum_{l=2}^{n} \binom{n}{l}\frac{B_{l}}{l+1}\beta _{n-l} (x\mid\lambda ). \end{aligned} \end{aligned}$$
Moreover,
$$\beta_{n}^{ (k )} (x\mid\lambda )=\sum _{l=0}^{n}\binom {n}{l}\beta_{l}^{ (k )} (\lambda ) (x\mid\lambda )_{n-l}. $$
By (2.4), we easily get
$$\begin{aligned} & \sum_{n=0}^{\infty}\beta_{n}^{ (k )} (x\mid\lambda )\frac{t^{n}}{n!} \\ & \quad =\frac{\operatorname{Li}_{k} (1-e^{-t} )}{ (1+\lambda t )^{\frac {1}{\lambda}}-1} (1+t )^{\frac{x}{\lambda}} \\ &\quad =\frac{t}{ (1+\lambda t )^{\frac{1}{\lambda}}-1} (1+t )^{\frac{x}{\lambda}}\frac{\operatorname{Li}_{k} (1-e^{-t} )}{t}. \end{aligned}$$
(2.6)
We observe that
$$\begin{aligned} \frac{1}{t}\operatorname{Li}_{k} \bigl(1-e^{-t} \bigr) & = \frac{1}{t}\sum_{n=1}^{\infty } \frac{1}{n^{k}} \bigl(1-e^{-t} \bigr)^{n} \\ & =\frac{1}{t}\sum_{n=1}^{\infty} \frac{ (-1 )^{n}}{n^{k}}n!\sum_{l=n}^{\infty}S_{2} (l,n )\frac{ (-t )^{l}}{l!} \\ & =\frac{1}{t}\sum_{l=1}^{\infty}\sum _{n=1}^{l}\frac{ (-1 )^{n+l}}{n^{k}}n!S_{2} (l,n )\frac{t^{l}}{l!} \\ & =\sum_{l=0}^{\infty}\sum _{n=1}^{l+1}\frac{ (-1 )^{n+l+1}}{n^{k}}n!\frac{S_{2} (l+1,n )}{l+1} \frac {t^{l}}{l!}. \end{aligned}$$
(2.7)
From (2.6) and (2.7), we have
$$\begin{aligned} & \sum_{n=0}^{\infty}\beta_{n}^{ (k )} (x\mid\lambda )\frac{t^{n}}{n!} \\ & \quad = \Biggl(\sum_{m=0}^{\infty} \beta_{m} (x\mid\lambda )\frac {t^{m}}{m!} \Biggr) \Biggl(\sum _{l=0}^{\infty} \Biggl(\sum _{p=1}^{l+1}\frac { (-1 )^{p+l+1}}{p^{k}}p!\frac{S_{2} (l+1,p )}{l+1} \Biggr)\frac{t^{l}}{l!} \Biggr) \\ & \quad =\sum_{n=0}^{\infty} \Biggl\{ \sum _{l=0}^{n}\binom{n}{l} \Biggl(\sum _{p=1}^{l+1}\frac{ (-1 )^{p+l+1}p!}{p^{k}}\frac{S_{2} (l+1,p )}{l+1} \Biggr)\beta_{n-l} (x\mid\lambda ) \Biggr\} \frac{t^{n}}{n!}. \end{aligned}$$
(2.8)

By comparing the coefficients on both sides of (2.8), we obtain the following theorem.

Theorem 2.2

For \(n\ge0\), we have
$$\beta_{n}^{ (k )} (x\mid\lambda )=\sum _{l=0}^{n}\binom {n}{l} \Biggl(\sum _{p=1}^{l+1}\frac{ (-1 )^{p+l+1}p!}{p^{k}}\frac {S_{2} (l+1,p )}{l+1} \Biggr)\beta_{n-l} (x\mid\lambda ). $$
It is easy to show that
$$\begin{aligned} & \frac{\operatorname{Li}_{k} (1-e^{-t} )}{ (1+\lambda t )^{\frac{1}{\lambda}}-1} (1+\lambda t )^{\frac {x+1}{\lambda}}-\frac{\operatorname{Li}_{k} (1-e^{-t} )}{ (1+\lambda t )^{\frac{1}{\lambda}}-1} (1+\lambda t )^{\frac {x}{\lambda}} \\ & \quad = (1+\lambda t )^{\frac{x}{\lambda}} \operatorname{Li}_{k} \bigl(1-e^{-t} \bigr) \\ &\quad = \Biggl(\sum_{l=0}^{\infty} (x\mid\lambda )_{l}\frac {t^{l}}{l!} \Biggr) \Biggl(\sum _{m=1}^{\infty}\frac{ (1-e^{-t} )^{m}}{m^{k}} \Biggr) \\ &\quad = \Biggl(\sum_{l=0}^{\infty} (x\mid\lambda )_{l}\frac {t^{l}}{l!} \Biggr) \Biggl(\sum _{m=0}^{\infty}\frac{ (1-e^{-t} )^{m+1}}{ (m+1 )^{k}} \Biggr) \\ &\quad = \Biggl(\sum_{l=0}^{\infty} (x\mid\lambda )_{l}\frac {t^{l}}{l!} \Biggr) \Biggl(\sum _{p=1}^{\infty} \Biggl(\sum_{m=0}^{p-1} \frac { (-1 )^{m+p+1}}{ (m+1 )^{k}} (m+1 )!S_{2} (p,m+1 ) \Biggr)\frac{t^{p}}{p!} \Biggr) \\ & \quad =\sum_{n=1}^{\infty} \Biggl\{ \sum _{p=1}^{n}\sum_{m=0}^{p-1} \frac{ (-1 )^{m+p+1}}{ (m+1 )^{k}} (m+1 )!S_{2} (p,m+1 )\binom{n}{p} (x\mid\lambda )_{n-p} \Biggr\} \frac {t^{n}}{n!}. \end{aligned}$$
(2.9)
On the other hand,
$$\begin{aligned} & \frac{\operatorname{Li}_{k} (1-e^{-t} )}{ (1+\lambda t )^{\frac{1}{\lambda}}-1} (1+\lambda t )^{\frac {x+1}{\lambda}}-\frac{\operatorname{Li}_{k} (1-e^{-t} )}{ (1+\lambda t )^{\frac{1}{\lambda}}-1} (1+\lambda t )^{\frac {x}{\lambda}} \\ &\quad =\sum_{n=0}^{\infty} \bigl\{ \beta_{n}^{ (k )} (x+1\mid \lambda )-\beta_{n}^{ (k )} (x\mid\lambda ) \bigr\} \frac{t^{n}}{n!}. \end{aligned}$$
(2.10)

Therefore, by (2.9) and (2.10), we obtain the following theorem.

Theorem 2.3

For \(n\ge1\), we have
$$\begin{aligned} & \beta_{n}^{ (k )} (x+1\mid\lambda )-\beta_{n}^{ (k )} (x\mid\lambda ) \\ & \quad =\sum_{p=1}^{n} \Biggl(\sum _{m=0}^{p-1}\frac{ (-1 )^{m+k+1}}{ (m+1 )^{k}} (m+1 )!S_{2} (k+m+1 ) \Biggr)\binom{n}{p} (x\mid\lambda )_{n-p}. \end{aligned}$$
Now, we note that
$$\begin{aligned} & \frac{\operatorname{Li}_{k} (1-e^{-t} )}{ (1+\lambda t )^{\frac{1}{\lambda}}-1} (1+\lambda t )^{\frac {x}{\lambda}} \\ & \quad =\frac{\operatorname{Li}_{k} (1-e^{-t} )}{ (1+\lambda t )^{\frac {d}{\lambda}}-1}\sum_{a=0}^{d-1} (1+ \lambda t )^{\frac {l+x}{\lambda}} \\ &\quad = \biggl(\frac{\operatorname{Li}_{k} (1-e^{-t} )}{t} \biggr)\frac{1}{d}\sum _{a=0}^{d-1}\frac{dt}{ (1+\lambda t )^{\frac{d}{\lambda }}-1} (1+\lambda t )^{\frac{l+x}{\lambda}} \\ & \quad =\sum_{l=0}^{\infty} \Biggl(\sum _{p=1}^{l+1}\frac{ (-1 )^{p+l+1}}{p^{k}}p!\frac{S_{2} (l+1,p )}{l+1} \Biggr)\frac {t^{l}}{l!} \\ & \qquad {}\times\sum_{a=0}^{d-1}\sum _{m=0}^{\infty}\beta _{m} \biggl( \frac{l+x}{d}\,\Big|\,\frac{\lambda}{d} \biggr)d^{m-1} \frac {t^{m}}{m!} \\ &\quad =\sum_{a=0}^{d-1} \Biggl(\sum _{n=0}^{\infty} \Biggl(\sum_{l=0}^{n} \sum_{p=1}^{l+1}\binom{n}{l} \frac{ (-1 )^{p+l+1}}{p^{k}}p!\frac {S_{2} (l+1,p )}{l+1}\beta_{n-l} \biggl( \frac{l+x}{d}\,\Big|\, \frac{\lambda}{d} \biggr)d^{n-l-1} \Biggr) \frac{t^{n}}{n!} \Biggr) \\ &\quad =\sum_{n=0}^{\infty} \Biggl\{ \sum _{a=0}^{d-1}\sum_{l=0}^{n} \sum_{p=1}^{l+1}\binom{n}{l} \frac{ (-1 )^{p+l+1}}{p^{k}}p!\frac {S_{2} (l+1,p )}{l+1}\beta_{n-l} \biggl( \frac{l+x}{d}\,\Big|\, \frac{\lambda}{d} \biggr)d^{n-l-1} \Biggr\} \frac{t^{n}}{n!}, \end{aligned}$$
(2.11)
where d is a fixed positive integer.
On the other hand,
$$\begin{aligned} & \frac{\operatorname{Li}_{k} (1-e^{-t} )}{ (1+\lambda t )^{\frac{1}{\lambda}}-1} (1+\lambda t )^{\frac {x}{\lambda}} \\ & \quad =\sum_{n=0}^{\infty}\beta_{n}^{ (k )} (x\mid\lambda )\frac{t^{n}}{n!}. \end{aligned}$$
(2.12)

Therefore, by (2.11) and (2.12), we obtain the following theorem.

Theorem 2.4

For \(n\ge0\), \(d\in\mathbb{N}\) and \(k\in\mathbb{Z}\), we have
$$\begin{aligned} & \beta_{n}^{ (k )} (x\mid\lambda ) \\ &\quad =\sum_{a=0}^{d-1}\sum _{l=0}^{n}\sum_{p=1}^{l+1} \binom{n}{l}\frac { (-1 )^{p+l+1}}{p^{k}}p!\frac{S_{2} (l+1,p )}{l+1}\beta_{n-l} \biggl(\frac{l+x}{d}\,\Big|\, \frac{\lambda }{d} \biggr)d^{n-l-1}. \end{aligned}$$
From (2.4), we can derive the following equation:
$$\begin{aligned} & \sum_{n=0}^{\infty}\beta_{n}^{ (k )} (x+y\mid\lambda )\frac{t^{n}}{n!} \\ & \quad =\frac{\operatorname{Li}_{k} (1-e^{-t} )}{ (1+\lambda t )^{\frac {1}{\lambda}}-1} (1+\lambda t )^{\frac{x+y}{\lambda}} \\ &\quad = \biggl(\frac{\operatorname{Li}_{k} (1-e^{-t} )}{ (1+\lambda t )^{\frac{1}{\lambda}}-1} (1+t\lambda )^{\frac{x}{\lambda }} \biggr) (1+\lambda t )^{\frac{y}{\lambda}} \\ & \quad = \Biggl(\sum_{l=0}^{\infty} \beta_{l}^{ (k )} (x\mid \lambda )\frac{t^{l}}{l!} \Biggr) \Biggl(\sum_{m=0}^{\infty} (y\mid\lambda )_{m}\frac{t^{m}}{m!} \Biggr) \\ & \quad =\sum_{n=0}^{\infty} \Biggl(\sum _{l=0}^{n}\binom{n}{l}\beta_{l}^{ (k )} (x\mid\lambda ) (y\mid\lambda )_{n-l} \Biggr)\frac{t^{n}}{n!}. \end{aligned}$$
(2.13)

Therefore, by (2.13), we obtain the following theorem.

Theorem 2.5

For \(n\ge0\), we have
$$\beta_{n}^{ (k )} (x+y\mid\lambda )=\sum _{l=0}^{n}\binom{n}{l}\beta_{l}^{ (k )} (x\mid\lambda ) (y\mid\lambda )_{n-l}. $$

Remark

$$\begin{aligned} & \frac{d}{dx}\beta_{n}^{ (k )} (x\mid \lambda ) \\ & \quad =\frac{d}{dx}\sum_{l=0}^{n} \binom{n}{l}\beta_{n-l}^{ (k )} (\lambda ) (x\mid\lambda )_{l} \\ & \quad =\sum_{l=0}^{n}\binom{n}{l} \beta_{n-l}^{ (k )} (\lambda )\sum_{j=0}^{l-1} \frac{1}{x-\lambda j}\prod_{i=0}^{l-1} (x-\lambda i ) \\ &\quad =\sum_{l=0}^{n}\binom{n}{l} \beta_{n-l}^{ (k )} (\lambda )\sum_{j=0}^{l-1} \prod_{\substack{i=0\\ i\neq j } }^{l-1} (x-\lambda i ). \end{aligned}$$

Declarations

Acknowledgements

The authors would like to thank the referees for their valuable comments.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Sogang University, Seoul, Republic of Korea
(2)
Department of Mathematics, Kwangwoon University, Seoul, Republic of Korea

References

  1. Carlitz, L: Degenerate Stirling, Bernoulli and Eulerian numbers. Util. Math. 15, 51-88 (1979) MATHGoogle Scholar
  2. Kim, DS, Kim, T, Dolgy, DV, Komatsu, T: Barnes-type degenerate Bernoulli polynomials. Adv. Stud. Contemp. Math. 25(1), 121-146 (2015) Google Scholar
  3. Kim, T: Barnes’ type multiple degenerate Bernoulli and Euler polynomials. Appl. Math. Comput. 258, 556-564 (2015) MathSciNetView ArticleGoogle Scholar
  4. Jolany, H, Mohebbi, H: Some results on Generalized multi poly-Bernoulli and Euler polynomials. Int. J. Math. Comb. 2, 117-129 (2011) Google Scholar
  5. Kim, DS, Kim, T, Mansour, T, Dolgy, DV: On poly-Bernoulli polynomials of the second kind with umbral calculus viewpoint. Adv. Differ. Equ. 2015(1), 27 (2015) MathSciNetView ArticleGoogle Scholar
  6. Kim, DS, Kim, T: Some identities of degenerate Euler polynomials arising from p-adic fermionic integrals on \(\Bbb{Z}_{p}\). Integral Transforms Spec. Funct. 26(4), 295-302 (2015) MathSciNetView ArticleGoogle Scholar
  7. Kim, DS, Kim, T: Higher-order Frobenius-Euler and poly-Bernoulli mixed-type polynomials. Adv. Differ. Equ. 2013, 251 (2013) View ArticleGoogle Scholar
  8. Kim, DS, Kim, T: Hermite and poly-Bernoulli mixed-type polynomials. Adv. Differ. Equ. 2013(343), 12 (2013) Google Scholar
  9. Kim, DS, Kim, T: A note on poly-Bernoulli and higher-order poly-Bernoulli polynomials. Russ. J. Math. Phys. 22(1), 26-33 (2015) MathSciNetView ArticleGoogle Scholar
  10. Kim, DS, Kim, T, Lee, SH: A note on poly-Bernoulli polynomials arising umbral calculus. Adv. Stud. Theor. Phys. 7(15), 731-744 (2013) Google Scholar
  11. Kim, T, Jang, YS, Seo, JJ: Poly-Bernoulli polynomials and their applications. Int. J. Math. Anal. 8(30), 1495-1503 (2014) Google Scholar
  12. Acikgoz, M, Erdal, D, Araci, S: A new approach to q-Bernoulli numbers and q-Bernoulli polynomials related to q-Bernstein polynomials. Adv. Differ. Equ. 9, Article ID 951764 (2010) MathSciNetView ArticleGoogle Scholar
  13. Araci, S, Acikgoz, M, Jolany, H: On the families of q-Euler polynomials and their applications. J. Egypt. Math. Soc. 23(1), 1-5 (2015) MathSciNetView ArticleGoogle Scholar
  14. Bayad, A, Hamahata, Y: Multiple polylogarithms and multi-poly-Bernoulli polynomials. Funct. Approx. Comment. Math. 46, 45-61 (2012) MathSciNetView ArticleMATHGoogle Scholar
  15. Bayad, A, Gaboury, S: Generalized Dirichlet L-function of arbitrary order with applications. Adv. Stud. Contemp. Math. (Kyungshang) 23(4), 607-619 (2013) MathSciNetMATHGoogle Scholar
  16. Gaboury, S, Tremblay, R, Fugère, B-J: Some explicit formulas for certain new classes of Bernoulli, Euler and Genocchi polynomials. Proc. Jangjeon Math. Soc. 17(1), 115-123 (2014) MathSciNetMATHGoogle Scholar
  17. Carlitz, L: A degenerate Staudt-Clausen theorem. Arch. Math. (Basel) 7, 28-33 (1956) MathSciNetView ArticleMATHGoogle Scholar
  18. Dere, R, Simsek, Y: Applications of umbral algebra to some special polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 22(3), 433-438 (2012) MathSciNetMATHGoogle Scholar
  19. Ding, D, Yang, J: Some identities related to the Apostol-Euler and Apostol-Bernoulli polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 20(1), 7-21 (2010) MathSciNetMATHGoogle Scholar
  20. Kim, T, Kwon, HK, Lee, SH, Seo, JJ: A note on poly-Bernoulli numbers and polynomials of the second kind. Adv. Differ. Equ. 2014, 219 (2014) MathSciNetView ArticleGoogle Scholar
  21. Kim, T: Some identities on the q-Euler polynomials of higher order and q-Stirling numbers by the fermionic p-adic integral on \(\Bbb{Z}_{p}\). Russ. J. Math. Phys. 16(4), 484-491 (2009) MathSciNetView ArticleMATHGoogle Scholar
  22. Luo, Q-M, Guo, B-N, Qi, F: On evaluation of Riemann zeta function \(\zeta(s)\). Adv. Stud. Contemp. Math. (Kyungshang) 7(2), 135-144 (2003) MathSciNetMATHGoogle Scholar
  23. Luo, Q-M, Qi, F: Relationships between generalized Bernoulli numbers and polynomials and generalized Euler numbers and polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 7(1), 11-18 (2003) MathSciNetMATHGoogle Scholar
  24. Park, J-W, Rim, S-H: On the modified q-Bernoulli polynomials with weight. Proc. Jangjeon Math. Soc. 17(2), 231-236 (2014) MathSciNetGoogle Scholar
  25. Şen, E: Theorems on Apostol-Euler polynomials of higher order arising from Euler basis. Adv. Stud. Contemp. Math. (Kyungshang) 23(2), 337-345 (2013) MathSciNetMATHGoogle Scholar
  26. Srivastava, HM, Kim, T, Simsek, Y: q-Bernoulli numbers and polynomials associated with multiple q-zeta functions and basic L-series. Russ. J. Math. Phys. 12(2), 241-268 (2005) MathSciNetMATHGoogle Scholar
  27. Zhang, Z, Yang, J: On sums of products of the degenerate Bernoulli numbers. Integral Transforms Spec. Funct. 20(9-10), 751-755 (2009) MathSciNetView ArticleMATHGoogle Scholar

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© Kim and Kim 2015

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