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A note on the Barnes-type q-Euler polynomials

Advances in Difference Equations20152015:250

https://doi.org/10.1186/s13662-015-0574-8

  • Received: 19 April 2015
  • Accepted: 13 July 2015
  • Published:

Abstract

In this paper, we consider the Barnes-type q-Euler polynomials which are derived from the fermionic p-adic q-integrals and investigate some identities of these polynomials. Furthermore, we define the Barnes-type q-Changhee polynomials and numbers, and we derive some identities related with the Barnes-type q-Euler polynomials and the Barnes-type q-Changhee polynomials.

Keywords

  • q-Euler polynomials
  • Barnes-type q-Euler polynomials
  • Changhee polynomials
  • Barnes-type q-Changhee polynomials
  • fermionic p-adic q-integral

MSC

  • 11B68
  • 11S40

1 Introduction

Let p be a fixed odd prime number. Throughout this paper, \(\mathbb{Z}_{p}\), \(\mathbb{Q}_{p}\), and \(\mathbb{C}_{p}\) will, respectively, denote the rings of p-adic integers, the fields of p-adic numbers, and the completion of the algebraic closure of \(\mathbb{Q}_{p}\). The p-adic norm \(|\cdot|_{p}\) is normalized as \(|p|_{p}=\frac{1}{p}\). The space of continuous functions on \(\mathbb{Z}_{p}\) is denoted by \(C(\mathbb{Z}_{p})\). Let q be an element in \(\mathbb{C}_{p}\) with \(|1-q|_{p} < p^{-\frac {1}{p-1}}\). The q-number of x is defined by \([x]_{q}=\frac{1-q^{x}}{1-q}\). For \(f\in C(\mathbb{Z}_{p})\), the fermionic p-adic integral on \(\mathbb {Z}_{p}\) is defined by Kim,
$$ I_{-q}(f)= \int_{\mathbb{Z}_{p}} f(x) \, d \mu_{-q}(x) =\lim_{N\rightarrow\infty} \frac{1}{[p^{N}]_{-q}} \sum _{x=0}^{p^{N}-1} f(x) (-q)^{x}\quad (\mbox{see [1--27]}), $$
(1)
where \([x]_{-q}= \frac{1-(-q)^{x}}{1+q}\). From (1), we note that
$$ q^{n} I_{-q}(f_{n})+(-1)^{n-1} I_{-q}(f)=[2]_{q} \sum_{l=0}^{n-1} (-1)^{n-1-l}q^{l} f(l), $$
(2)
where \(f_{n} (x)=f(x+n)\) (\(n\geq1\)). In particular, for \(n=1\),
$$ qI_{-q}(f_{1})+ I_{-q}(f)=[2]_{q} f(0). $$
(3)
We note that
$$ \int_{\mathbb{Z}_{p}} e^{(x+y)t}\, d \mu_{-q} (y)= \frac{[2]_{q}}{qe^{t}+1}e^{xt}. $$
(4)
As is well known, the q-Euler polynomials are defined by Kim,
$$ \frac{[2]_{q}}{qe^{t}+1} e^{xt} = \sum _{n=0}^{\infty}E_{n,q} (x) \frac{t^{n}}{n!} \quad (\mbox{see [2, 9, 22, 28]}). $$
(5)
When \(x=0\), \(E_{n,q}=E_{n,q}(0)\) are called the q-Euler numbers. We note that \(\lim_{q\rightarrow1} E_{n,q}(x)=E_{n}(x)\), where \(E_{n}(x)\) are called the Euler polynomials which are defined by the generating function,
$$ \frac{2}{e^{t}+1} e^{xt} = \sum_{n=0}^{\infty}E_{n} (x) \frac{t^{n}}{n!} . $$
The Stirling number of the first kind is given by the generating function,
$$ (x)_{m} =\sum_{l=0}^{m} S_{1}(m,l)x^{l} \quad (m\geq0) $$
(6)
and the Stirling number of the second kind is defined by the generating function,
$$ \bigl(e^{t}-1\bigr)^{m} =m!\sum _{l=m}^{\infty}S_{2}(l,m)\frac{t^{l}}{l!} \quad (m\geq0)\ (\mbox{see [7, 8, 15, 17]}). $$
(7)

In [21], Kim (2010) presented the generating functions related to the q-Euler polynomials of higher order and gave some interesting identities involving these polynomials. In [2], Bayad and Kim (2011) studied some relations involving values of q-Bernoulli, q-Euler, and Bernstein polynomials (see [36, 10, 20, 2225, 29, 30]). Recently, Kim et al. studied some identities for q-analogs of the Changhee polynomials (see [11, 15]), for various degenerate Bernoulli polynomials (see [13, 16, 17, 22]), and for q-analogs of the Boole polynomials (see [10, 12]).

In recent years, a lot of people have studied various types of q-Euler polynomials and obtained many results which are interesting in number theory and combinatorics. To cite a few, in [28] one obtained eight basic identities of symmetry in three variables related to the q-Euler polynomials and a q-analog of alternating power sums. The derivation is based on the p-adic q-integrals in our case but on the p-adic integrals in [28]. In [9], some combinatorial identities involving q-Euler numbers and polynomials were obtained by adopting the ideas from [25]. It is fascinating that very recently some degenerate versions of many important polynomials were studied and some interesting results were obtained including the degenerate q-Euler polynomials. The aim of this paper is to define Barnes-type q-Euler numbers and polynomials in terms of p-adic q-integrals and to derive Witt-type formulas for them. Further, we find the connection between Barnes-type q-Euler polynomials and Barnes-type Frobenius polynomials and Barnes-type q-Changhee polynomials. This generalizes the Euler polynomials introduced in [21] by Kim.

In a forthcoming paper, we would like to give some of the applications of our results to symmetric identities involving Barnes-type q-Euler numbers and polynomials, to derivation of many identities of combinatorial nature. Also, we will investigate further properties, recurrence relations, and combinatorial identities for the Barnes-type polynomials by utilizing umbral calculus and degenerate versions of them.

The main results of this paper are some identities of the Barnes-type q-Euler polynomials. Furthermore, we define the Barnes-type q-Changhee polynomials and numbers, and we derive some identities related with the Barnes-type q-Euler polynomials and the Barnes-type q-Changhee polynomials.

2 The Barnes-type q-Euler polynomials and numbers

Let \(w_{1}, \ldots, w_{r} \in\mathbb{C}_{p}\). The Barnes-type Euler polynomials are defined by the generating function
$$ \prod_{l=1}^{r} \biggl( \frac{2}{e^{w_{i}t}+1} \biggr) e^{xt} =\sum_{n=0}^{\infty}E_{n}(x|w_{1}, \ldots, w_{r}) \frac{t^{n}}{n!}. $$
(8)
When \(x=0\), \(E_{n}(w_{1}, \ldots, w_{r})= E_{n}(0|w_{1}, \ldots, w_{r})\) are called the Barnes-type Euler numbers (see [1, 3, 5, 6, 9, 12, 14, 2830]). By (4), we get
$$ \int_{\mathbb{Z}_{p}} \cdots \int_{\mathbb{Z}_{p}} e^{(w_{1}x_{1}+\cdots+w_{r}x_{r}+x)t} \, d\mu_{-q}(x_{1})\cdots\, d \mu_{-q} (x_{r}) =\prod_{l=1}^{r} \biggl(\frac{[2]_{q}}{qe^{w_{l}t}+1} \biggr) e^{xt}, $$
(9)
for \(|t|_{p} < p^{-\frac{1}{p-1}}\). From (9), the Barnes-type q-Euler polynomials are defined by the generating function,
$$ [2]_{q}^{r} \prod _{l=1}^{r} \biggl(\frac{1}{qe^{w_{1}t}+1} \biggr) e^{xt} =\sum_{n=0}^{\infty}E_{n,q}(x| w_{1},\ldots,w_{r})\frac{t^{n}}{n!}. $$
(10)
When \(x=0\), \(E_{n,q}(w_{1}, \ldots, w_{r})= E_{n,q}(0|w_{1}, \ldots, w_{r})\) are called the Barnes-type q-Euler numbers. By (9) and (10), we get
$$\begin{aligned}& \sum_{n=0}^{\infty}E_{n,q}(x|w_{1}, \ldots, w_{r}) \frac{t^{n}}{n!} \\& \quad = \int_{\mathbb{Z}_{p}} \cdots\int_{\mathbb{Z}_{p}}e^{(w_{1}x_{1}+\cdots+w_{r}x_{r}+x)t} \, d\mu_{-q}(x_{1})\cdots\, d\mu_{-q}(x_{r}) \\& \quad = \sum_{n=0}^{\infty}\int _{\mathbb{Z}_{p}} \cdots\int_{\mathbb{Z}_{p}} (w_{1}x_{1}+ \cdots+w_{r}x_{r}+x)^{n} \, d\mu_{-q}(x_{1}) \cdots\, d\mu_{-q}(x_{r}) \frac{t^{n}}{n!}. \end{aligned}$$
(11)
From (11), we obtain the following theorem.

Theorem 2.1

For \(n\geq0\), we have
$$ E_{n,q}(x| w_{1}, \ldots, w_{r}) = \int_{\mathbb{Z}_{p}} \cdots\int_{\mathbb{Z}_{p}} (w_{1}x_{1}+\cdots+w_{r}x_{r}+x)^{n} \, d\mu_{-q}(x_{1})\cdots\, d\mu_{-q}(x_{r}). $$
(12)
From (12), we note that
$$ E_{n,q}( w_{1}, \ldots, w_{r}) = \int_{\mathbb{Z}_{p}} \cdots\int_{\mathbb{Z}_{p}} (w_{1}x_{1}+\cdots+w_{r}x_{r})^{n} \, d\mu_{-q}(x_{1})\cdots\, d\mu_{-q}(x_{r}). $$
(13)
Now, we observe that
$$\begin{aligned} \sum_{n=0}^{\infty}E_{n,q}(x|w_{1}, \ldots, w_{r}) \frac{t^{n}}{n!} =& \frac{(1+q)^{r}}{(qe^{w_{1}t}+1)\cdots(qe^{w_{r}t}+1)}e^{xt} \\ =& \frac{(1+q^{-1})^{r}}{(e^{w_{1}t}+q^{-1})\cdots(e^{w_{r}t}+q^{-1})}e^{xt} \\ =& \sum_{n=0}^{\infty}H_{n}\bigl(x, -q^{-1}|w_{1}, \ldots,w_{r}\bigr) \frac{t^{n}}{n!}, \end{aligned}$$
(14)
where \(H_{n}(x,u|w_{1},\ldots,w_{r})\) are called the Barnes-type Frobenius-Euler polynomials defined by the generating function,
$$ \frac{(1-u)^{r}}{(e^{w_{1}t}-u)\cdots(e^{w_{r}t}-u)} e^{xt} =\sum _{n=0}^{\infty}H_{n}(x, u|w_{1}, \ldots, w_{r}) \frac{t^{n}}{n!}\quad (\mbox{see [2, 4]}). $$
(15)
Therefore, by (14), we obtain the following theorem.

Theorem 2.2

Let \(n\geq0\), we have
$$ E_{n,q}(x| w_{1}, \ldots, w_{r}) = H_{n} \bigl(x,-q^{-1}| w_{1},\ldots, w_{r}\bigr). $$
(16)
Let \(n\geq0\) and \(d \in\mathbb{N}\) with \(d\equiv1\ (\operatorname{mod} 2)\). By (2), we get
$$ q^{d} I_{-q} (f_{d})+I_{-q}(f)= [2]_{q} \sum_{l=0}^{d-1}(-q)^{l} f(l). $$
(17)
By (17), we get
$$\begin{aligned}& \sum_{n=0}^{\infty}E_{n,q}(x|w_{1}, \ldots, w_{r}) \frac{t^{n}}{n!} \\& \quad = \biggl(\frac{[2]_{q}}{[2]_{q^{d}}} \biggr)\int_{\mathbb{Z}_{p}}\cdots\int _{\mathbb{Z}_{p}} e^{(w_{1}x_{1}+\cdots+w_{r}x_{r}+x)t} \, d\mu_{-q} (x_{1})\cdots\, d\mu_{-q}(x_{r}) \\& \quad = \biggl(\frac{[2]_{q}}{[2]_{q^{d}}} \biggr)\prod_{l=1}^{r} \biggl( \frac {[2]_{q}}{q^{d}e^{w_{l}dt}+1} \biggr)\sum_{l_{1},\ldots, l_{r}=0}^{d-1} (-q)^{l_{1}+\cdots+l_{r}} e^{(w_{1}l_{1}+\cdots+w_{r}l_{r}+x)t} \\& \quad = \biggl(\frac{[2]_{q}}{[2]_{q^{d}}} \biggr) \Biggl( \sum _{m=0}^{\infty}E_{m,q^{d}}(dw_{1}, \ldots,dw_{r}) \frac{t^{m}}{m!} \Biggr) \\& \qquad {}\times\sum _{l_{1},\ldots, l_{r}=0}^{d-1} (-q)^{l_{1}+\cdots+l_{r}} \sum _{k=0}^{\infty}(w_{1}l_{1}+\cdots +w_{r}l_{r}+x)^{k}\frac{t^{k}}{k!} \\& \quad = \sum_{n=0}^{\infty}\biggl( \frac{[2]_{q}}{[2]_{q^{d}}} \biggr) \Biggl( \sum_{k=0}^{n} \binom{n}{k} \sum_{l_{1},\ldots, l_{r}=0}^{d-1} (-q)^{l_{1}+\cdots +l_{r}} (w_{1}l_{1}+\cdots+w_{r}l_{r}+x)^{k} \\& \qquad {}\times E_{n-k,q^{d}}(dw_{1}, \ldots,dw_{r}) \Biggr) \frac{t^{n}}{n!}. \end{aligned}$$
(18)
Thus, by (18), we obtain the following theorem.

Theorem 2.3

Let \(n\geq0\). Then, for positive integer d with \(d\equiv1\ (\operatorname{mod} 2)\),
$$\begin{aligned}& E_{n,q}( x| w_{1}, \ldots, w_{r}) \\& \quad = \biggl(\frac{[2]_{q}}{[2]_{q^{d}}} \biggr) \sum_{k=0}^{n} \binom{n}{k} \sum_{l_{1}, \ldots, l_{r}=0}^{d-1} (-q)^{l_{1}+\cdots+l_{r}}(w_{1}l_{1}+\cdots+ w_{1}l_{1}+x)^{k} \\& \qquad {}\times E_{n-k,q^{d}}(dw_{1}, \ldots, dw_{r}). \end{aligned}$$
(19)
We note that in [6], the authors considered the q-extensions of Changhee polynomials which are derived from the fermionic p-adic q-integral on \(\mathbb {Z}_{p}\), and they gave some identities for these polynomials. Finally, we consider the Barnes-type q-Changhee polynomials. By (3), we note that, for \(l=1, \ldots,r\),
$$ \int_{\mathbb{Z}_{p}}(1+t)^{w_{l}x}\, d \mu_{-q}(x)= \frac{[2]_{q}}{q(1+t)^{w_{l}}+1}, $$
(20)
where \(|t|_{p} < p^{-\frac{1}{p-1}}\). By (20), we get
$$ \int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}(1+t)^{w_{1}x_{1}+\cdots +w_{r}x_{r}+x} \, d\mu_{-q}(x_{1})\cdots \, d \mu_{-q}(x_{r}) = \prod_{l=1}^{r} \frac{[2]_{q}}{q(1+t)^{w_{l}}+1}(1+t)^{x}. $$
(21)
From (21), the Barnes-type q-Changhee polynomials are defined by the generating function,
$$ \prod_{l=1}^{r} \frac{[2]_{q}}{q(1+t)^{w_{l}}+1}(1+t)^{x} = \sum_{n=0}^{\infty}\mathit{Ch}_{n,q}(x|w_{1}, \ldots, w_{r}) \frac{t^{n}}{n!} . $$
(22)
When \(x=0\), \(\mathit{Ch}_{n,q}(w_{1}, \ldots, w_{r})=\mathit{Ch}_{n,q}(0|w_{1}, \ldots, w_{r})\) are called the Barnes-type q-Changhee numbers (see [7, 11, 13, 15]). By (21) and (22), we have
$$\begin{aligned}& \sum_{n=0}^{\infty}\mathit{Ch}_{n,q}(x|w_{1}, \ldots, w_{r}) \frac{t^{n}}{n!} \\& \quad = \int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}}(1+t)^{w_{1}x_{1}+\cdots +w_{r}x_{r}+x} \, d\mu_{-q}(x_{1})\cdots\, d \mu_{-q}(x_{r}) \\& \quad = \sum_{n=0}^{\infty}\int _{\mathbb{Z}_{p}}\cdots\int_{\mathbb {Z}_{p}}\binom{w_{1}x_{1}+\cdots+w_{r}x_{r}+x}{n} t^{n} \, d\mu_{-q}(x_{1})\cdots \, d \mu _{-q}(x_{r}) \\& \quad = \sum_{n=0}^{\infty}\int _{\mathbb{Z}_{p}}\cdots\int_{\mathbb {Z}_{p}}(w_{1}x_{1}+ \cdots+w_{r}x_{r}+x)_{n} \, d\mu_{-q}(x_{1}) \cdots \, d \mu_{-q}(x_{r}) \frac{t^{n}}{n!} \\& \quad = \sum_{n=0}^{\infty}\sum _{l=0}^{n} \int_{\mathbb{Z}_{p}}\cdots\int _{\mathbb{Z}_{p}}S_{1}(n,l) (w_{1}x_{1}+ \cdots+w_{r}x_{r}+x)^{l}\, d\mu_{-q}(x_{1}) \cdots \, d \mu_{-q}(x_{r})\frac{t^{n}}{n!} \\& \quad = \sum_{n=0}^{\infty}\sum _{l=0}^{n} S_{1}(n,l) E_{l,q}(x|w_{1}, \ldots, w_{r}) \frac{t^{n}}{n!}. \end{aligned}$$
(23)
By (23), we obtain the following theorem.

Theorem 2.4

Let \(n\geq0\). Then we have
$$ \mathit{Ch}_{n,q}(x|w_{1}, \ldots, w_{r}) = \sum_{l=0}^{n} S_{1}(n,l) E_{l,q}(x|w_{1}, \ldots, w_{r}) . $$
(24)
By replacing t by \(e^{t}-1\), we have
$$\begin{aligned} \prod_{l=1}^{r} \frac{[2]_{q}}{qe^{w_{l}t}+1}e^{xt} =& \sum_{m=0}^{\infty}\mathit{Ch}_{m,q}(x|w_{1}, \ldots, w_{r}) \frac{(e^{t}-1)^{m}}{m!} \\ =& \sum_{m=0}^{\infty}\mathit{Ch}_{m,q}(x|w_{1}, \ldots, w_{r}) \frac{1}{m!} m! \sum_{n=m}^{\infty}S_{2}(n,m) \frac{t^{m}}{m!} \\ =& \sum_{n=0}^{\infty}\sum _{m=0}^{n} S_{2}(n,m) \mathit{Ch}_{m,q}(x|w_{1}, \ldots, w_{r}) \frac{t^{n}}{n!}. \end{aligned}$$
(25)
By (25) we obtain the following theorem.

Theorem 2.5

Let \(n\geq0\). Then we have
$$ E_{n,q}(x|w_{1}, \ldots, w_{r}) = \sum_{m=0}^{n} S_{2}(n,m) \mathit{Ch}_{n,q}(x|w_{1}, \ldots, w_{r}) . $$
(26)

Declarations

Acknowledgements

The authors would like to thank the referees and Professor Ravi P. Agarwal for providing very valuable comments and suggestions. This paper was supported by Wonkwang University in 2013.

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)
Graduate School of Education, Konkuk University, Seoul, 143-701, Republic of Korea
(2)
Division of Mathematics and Informational Statistics, and Nanoscale Science and Technology Institute, Wonkwang University, Iksan, 570-749, Republic of Korea

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