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 [3–6, 10, 20, 22–25, 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, 28–30]). 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)