Open Access

Some identities of higher order Barnes-type q-Bernoulli polynomials and higher order Barnes-type q-Euler polynomials

Advances in Difference Equations20152015:162

https://doi.org/10.1186/s13662-015-0495-6

Received: 11 March 2015

Accepted: 6 May 2015

Published: 29 May 2015

Abstract

In this paper, we consider higher order Barnes-type q-Bernoulli polynomials and numbers and investigate some identities of them. Furthermore, we discuss some identities of higher order Barnes-type q-Euler polynomials and numbers.

Keywords

p-adic invariant integral Bernoulli polynomials Euler polynomials higher order Barnes-type q-Bernoulli polynomials and numbers

MSC

11B68 11S40

1 Introduction

Let p be a given odd prime number. Throughout this paper, we assume that \(\mathbb{Z}_{p}\), \(\mathbb{Q}_{p}\) and \(\mathbb{C}_{p}\) will, respectively, denote the rings of p-adic integers, the fields of p-adic numbers and the completion of algebraic closure of \(\mathbb{Q}-p\). The p-adic norm \(|p|_{p}=\frac{1}{p}\). Let \(\operatorname{UD}(\mathbb{Z}_{p})\) be the space of uniformly differentiable functions on \(\mathbb{Z}_{p}\). For \(f\in \operatorname{UD}(\mathbb{Z}_{p})\), the bosonic p-adic integral on \(\mathbb {Z}_{p}\) is defined as
$$ I_{0}(f)= \int_{\mathbb{Z}_{p}} f(x)\, d \mu_{0}(x) =\lim_{N\rightarrow\infty} \frac{1}{p^{N}} \sum _{x=0}^{p^{N}-1} f(x) \quad (\mbox{see [1--12]}). $$
(1)
It is well known that an integral equation of the bosonic p-adic integral \(I_{0}\) on \(\mathbb{Z}_{p}\),
$$ I_{0}(f_{1})-I_{0}(f)=f'(0), $$
(2)
where \(f_{1}(x)=f(x+1)\). Higher order Bernoulli polynomials are defined by Kim to be
$$ \biggl( \frac{t}{e^{t}-1} \biggr)^{r} e^{xt} = \sum_{n=0}^{\infty}B_{n}^{(r)} (x) \frac{t^{n}}{n!}\quad (\mbox{see [5, 13--16]}). $$
(3)
When \(x=0\), \(B_{n}^{(r)}=B_{n}^{(r)}(0)\) is called higher order Bernoulli numbers. Higher order Barnes-type Bernoulli polynomials are defined by Kim to be
$$ \prod_{i=1}^{r} \biggl( \frac{t}{e^{a_{i}t}-1} \biggr)^{r} e^{xt} = \sum _{n=0}^{\infty}B_{n}^{(r)} (x|a_{1}, \ldots, a_{r}) \frac{t^{n}}{n!}\quad (\mbox{see [11--15, 17--21]}). $$
(4)
When \(x=0\), \(B_{n}^{(r)}(a_{1}, \ldots, a_{r})=B_{n}^{(r)}(0|a_{1}, \ldots, a_{r})\) is called higher order Barnes-type Bernoulli numbers.

In this paper we consider higher order Barnes-type q-Bernoulli polynomials and numbers and investigate some identities of them. We also discuss some identities of higher order Barnes-type q-Euler polynomials and numbers.

2 Higher order Barnes-type q-Bernoulli polynomials and numbers

In this section, we assume that \(q\in\mathbb{C}_{p}\) with \(|1-q|_{p}< p^{-\frac{1}{p-1}}\). By (2), if we take \(f(x)=q^{y} e^{(x+y)t}\), then we get
$$ \int_{\mathbb{Z}_{p}} q^{y} e^{(x+y)t} \, d \mu_{0} (y)= \frac{t+ \log q}{qe^{t}-1}e^{xt}, $$
(5)
where \(f_{1}(x)=f(x+1)\). q-Bernoulli polynomials are defined by Kim to be
$$ \frac{t+ \log q}{qe^{t}-1}e^{xt} = \sum _{n=0}^{\infty}B_{n,q}(x) \frac {t^{n}}{n!}\quad (\mbox{see [13--15, 17, 19--21]}). $$
(6)
When \(x=0\), \(B_{n,q}=B_{n,q}(0)\) is called q-Bernoulli numbers.
Higher order q-Bernoulli polynomials are defined as
$$ \biggl( \frac{t+ \log q}{qe^{t}-1} \biggr)^{r} e^{xt} = \sum_{n=0}^{\infty}B_{n,q}^{(r)}(x) \frac{t^{n}}{n!}. $$
(7)
When \(x=0\), \(B_{n,q}^{(r)}=B_{n,q}^{(r)}(0)\) is called higher order q-Bernoulli numbers.
We define higher order Barnes-type q-Bernoulli polynomials as follows:
$$ \frac{(t+ \log q)^{r}}{ ( q^{a_{1}}e^{a_{1}t}-1 )\cdots ( q^{a_{r}}e^{a_{r}t}-1 )} e^{xt} = \sum _{n=0}^{\infty}B_{n,q}(x|a_{1}, \ldots, a_{r}) \frac{t^{n}}{n!}. $$
(8)
When \(x=0\), \(B_{n,q}(a_{1}, \ldots, a_{r})= B_{n,q}(0|a_{1}, \ldots, a_{r})\) is called higher order Barnes-type q-Bernoulli numbers. By (5), we get
$$\begin{aligned}& \int_{\mathbb{Z}_{p}} q^{a_{1}x_{1}+\cdots+a_{r}t_{r}} e^{(a_{1}x_{1}+\cdots +a_{r}x_{r} +x)t} \, d\mu_{0}(x_{1})\cdots \, d\mu_{0} (x_{r}) \\& \quad = \Biggl( \prod_{i=1}^{r} a_{i} \Biggr) \frac{(t+ \log q)^{r}}{ ( q^{a_{1}}e^{a_{1}t}-1 )\cdots ( q^{a_{r}}e^{a_{r}t}-1 )} e^{xt}. \end{aligned}$$
(9)
By (9) and (8), we get
$$\begin{aligned}& \sum_{n=0}^{\infty}B_{n,q}(x|a_{1}, \ldots, a_{r}) \frac{t^{n}}{n!} \\& \quad =\frac{(t+ \log q)^{r}}{ ( q^{a_{1}}e^{a_{1}t}-1 )\cdots ( q^{a_{r}}e^{a_{r}t}-1 )} e^{xt} \\& \quad = \Biggl( \prod_{i=1}^{r} a_{i} \Biggr)^{-1} \int_{\mathbb{Z}_{p}} \cdots\int _{\mathbb{Z}_{p}} q^{a_{1}x_{1}+\cdots +a_{r}x_{r}}e^{(a_{1}x_{1}+\cdots+a_{r}x_{r}+x)t} \, d \mu_{0}(x_{1})\cdots\, d\mu_{0}(x_{r}) \\& \quad = \sum_{n=0}^{\infty}\Biggl( \Biggl( \prod_{i=1}^{r} a_{i} \Biggr)^{-1} \int_{\mathbb{Z}_{p}} \cdots\int _{\mathbb{Z}_{p}} q^{a_{1}x_{1}+\cdots+a_{r}x_{r}+x} \\& \qquad {}\times(a_{1}x_{1}+ \cdots+a_{r}x_{r}+x)^{n} \, d\mu_{0}(x_{1}) \cdots\, d\mu_{0}(x_{r}) \Biggr) \frac{t^{n}}{n!}. \end{aligned}$$
(10)

From (10), we obtain the following theorem.

Theorem 2.1

Let \(n\in\mathbb{N}\cup\{0\}\). Then we have
$$\begin{aligned}& B_{n,q}(x| a_{1}, \ldots, a_{r}) \\& \quad = \Biggl( \prod_{i=1}^{r} a_{i} \Biggr)^{-1} \int_{\mathbb{Z}_{p}} \cdots\int _{\mathbb{Z}_{p}} q^{a_{1}x_{1}+\cdots+a_{r}x_{r}+x} \\& \qquad {}\times(a_{1}x_{1}+ \cdots+a_{r}x_{r}+x)^{n} \, d\mu_{0}(x_{1}) \cdots\, d\mu_{0}(x_{r}). \end{aligned}$$
(11)
From (1), we have
$$\begin{aligned} \int_{\mathbb{Z}_{p}} f(x)\, d \mu_{0}(x) =& \lim_{N\rightarrow\infty} \frac{1}{p^{N}} \sum_{x=0}^{dp^{N}-1} f(x) \\ =& \frac{1}{d} \lim_{N\rightarrow\infty} \frac{1}{p^{N}} \sum _{a=0}^{d-1} \sum _{x=0}^{p^{N}-1} f(a+dx) \\ =& \frac{1}{d}\sum_{a=0}^{d-1} \int _{\mathbb{Z}_{p}} f(a+dx)\, d \mu_{0}(x). \end{aligned}$$
(12)
By (12), we have
$$\begin{aligned}& \Biggl( \prod_{i=1}^{r} a_{i} \Biggr)^{-1}\int_{\mathbb{Z}_{p}} \cdots\int _{\mathbb{Z}_{p}} q^{a_{1} x_{1}+\cdots+a_{r} x_{r}} e^{(a_{1}d x_{1}+\cdots+a_{r}d x_{r}+x)t} \, d \mu_{0}(x_{1})\cdots\, d\mu_{0}(x_{r}) \\& \quad = \frac{1}{d^{r}}\sum_{l_{1}, \ldots, l_{r}=0}^{d-1} q^{a_{1}l_{1}\cdots+a_{r}l_{r}} \Biggl( \prod_{i=1}^{r} a_{i} \Biggr)^{-1} \int_{\mathbb{Z}_{p}}\cdots\int _{\mathbb{Z}_{p}} q^{a_{1}\, dx_{1}+\cdots+a_{r}\, dx_{r}} \\& \qquad {}\times e^{(a_{1}l_{1}\cdots+a_{r}l_{r}+ x+ a_{1}\, dx_{1}+\cdots+a_{r}\, dx_{r})t} \, d \mu_{0}(x_{1})\cdots\, d\mu_{0}(x_{r}) \\& \quad = \frac{1}{d^{r}}\sum_{l_{1}, \ldots, l_{r}=0}^{d-1} q^{a_{1}l_{1}\cdots+a_{r}l_{r}} \Biggl( \prod_{i=1}^{r} a_{i} \Biggr)^{-1}\int_{\mathbb{Z}_{p}}\cdots\int _{\mathbb{Z}_{p}} q^{a_{1}\, dx_{1}+\cdots+a_{r}\, dx_{r}} \\& \qquad {}\times e^{ ( \frac{a_{1}l_{1}+\cdots+a_{r}l_{r} +x}{d} + a_{1}x_{1}+\cdots+a_{r} x_{r} ) \, dt}\, d \mu_{0}(x_{1})\cdots\, d\mu_{0}(x_{r}) \\& \quad = \sum_{l_{1}, \ldots, l_{r}=0}^{d-1}q^{a_{1}l_{1}\cdots+a_{r}l_{r}} \sum_{n=0}^{\infty}\frac{d^{n}}{d^{r}} \Biggl( \prod_{i=1}^{r} a_{i} \Biggr)^{-1} \int_{\mathbb{Z}_{p}}\cdots\int _{\mathbb{Z}_{p}} q^{a_{1}x_{1}+\cdots+a_{r}x_{r}} \\& \qquad {}\times \biggl( \frac{a_{1}l_{1}+\cdots+a_{r}l_{r} +x}{d} + a_{1}x_{1}+ \cdots+a_{r} x_{r} \biggr)^{n} \, d \mu_{0}(x_{1})\cdots\, d\mu_{0}(x_{r}) \frac{t^{n}}{n!} \\& \quad = \sum_{n=0}^{\infty}d^{n-r} \sum_{l_{1}, \ldots, l_{r}=0}^{d-1} q^{a_{1}x_{1}+\cdots+a_{r}x_{r}} B_{n,q} \biggl( \frac{a_{1}l_{1}+\cdots+a_{r}l_{r} +x}{d} \Big| a_{1}, \ldots, a_{r} \biggr) \frac{t^{n}}{n!}. \end{aligned}$$
(13)

By (8), (9), (11) and (13), we obtain the following theorem.

Theorem 2.2

Let \(n\in\mathbb{N}\cup\{0\}\). Then we have
$$\begin{aligned}& B_{n,q}(x| a_{1}, \ldots, a_{r}) \\& \quad = d^{n-r} \sum_{l_{1}, \ldots, l_{r}=0}^{d-1}q^{a_{1}x_{1}+\cdots+a_{r}x_{r}} B_{n,q} \biggl( \frac{l_{1}x_{1}+\cdots+l_{r}x_{r} +x}{d} \Big| a_{1}, \ldots, a_{r} \biggr). \end{aligned}$$
(14)
It is well known that an integral equation of the bosonic p-adic integral \(I_{0}\) on \(\mathbb{Z}_{p}\) satisfies the following integral equation:
$$ I_{0}(f_{n})-I_{0}(f)= \sum _{i=1}^{n-1} f'(i). $$
(15)
If we take \(f(x_{i})=q^{a_{i}x_{i}}e^{a_{i}x_{i}t}\) for \(i=1, \ldots, r\), then we have
$$ \int_{\mathbb{Z}_{p}} q^{a_{i} x_{i}} e^{a_{i}x_{i} t}\, d\mu_{0}(x_{i}) =\frac{a_{i}(t+\log q)}{q^{a_{i}n}e^{a_{i} nt}-1}\sum _{l=0}^{n-1} q^{a_{i} l}e^{a_{i}lt}. $$
(16)
By (16), we get
$$\begin{aligned}& \Biggl( \prod_{i=1}^{r} a_{i} \Biggr)^{-1} \int_{\mathbb{Z}_{p}} \cdots \int _{\mathbb{Z}_{p}} q^{a_{1}x_{1}+\cdots+a_{r}x_{r}} e^{(a_{1}x_{1}+\cdots+a_{r}x_{r}) t}\, d \mu_{0}(x_{1}) \cdots \, d\mu_{0}(x_{r}) \\& \quad = \frac{(t+\log q)^{r}}{ (q^{a_{1}n}e^{a_{1} nt}-1 ) \cdots (q^{a_{r}n}e^{a_{r} nt}-1 )} \sum_{l_{1}, \ldots, l_{r}=0}^{n-1} q^{a_{1}l_{1}+\cdots+a_{r}l_{r}} e^{(a_{1}l_{1}+\cdots+a_{r}l_{r}) t} \\& \quad = \Biggl( \sum_{k=0}^{\infty}B_{k,q}(na_{1}, \ldots, na_{r}) \frac {t^{k}}{k!} \Biggr)\sum_{l_{1}, \ldots, l_{r}=0}^{n-1} q^{a_{1}l_{1}+\cdots+a_{r}l_{r}} \sum _{j=0}^{\infty}(a_{1}l_{1}+ \cdots+ a_{r}l_{r})^{j} \frac{t^{j}}{j!} \\& \quad = \sum_{l_{1}, \ldots, l_{r}=0}^{n-1} \sum _{k=0}^{\infty}\sum_{j=0}^{\infty}q^{a_{1}l_{1}+\cdots+a_{r}l_{r}} (a_{1}l_{1}+\cdots+ a_{r}l_{r})^{j} B_{k,q}(na_{1}, \ldots, na_{r})\frac{t^{k+j}}{k!j!} \\& \quad = \sum_{m=0}^{\infty}\sum _{l_{1}, \ldots, l_{r}=0}^{n-1} \sum_{j=0}^{m} \binom{m}{j} q^{a_{1}l_{1}+\cdots+a_{r}l_{r}}(a_{1}l_{1}+\cdots+ a_{r}l_{r})^{j} \\& \qquad {}\times B_{m-j,q}(na_{1}, \ldots, na_{r}) \frac{t^{m}}{m!}. \end{aligned}$$
(17)

Thus, by (11) and (17), we obtain the following theorem.

Theorem 2.3

Let \(n\in\mathbb{N}\cup\{0\}\). Then we have
$$\begin{aligned}& B_{n,q}( a_{1}, \ldots, a_{r}) \\& \quad = \sum_{l_{1}, \ldots, l_{r}=0}^{n-1} \sum _{j=0}^{m} \binom{m}{j} q^{a_{1}l_{1}+\cdots+a_{r}l_{r}}(a_{1}l_{1}+ \cdots+ a_{r}l_{r})^{j} B_{m-j,q}(na_{1}, \ldots, na_{r}). \end{aligned}$$
(18)
By (16), we get
$$\begin{aligned}& \Biggl( \prod_{i=1}^{r} a_{i} \Biggr)^{-1} \int_{\mathbb{Z}_{p}} \cdots \int _{\mathbb{Z}_{p}} q^{a_{1}x_{1}+\cdots+a_{r}x_{r}} e^{(a_{1}x_{1}+\cdots+a_{r}x_{r}) t}\, d \mu_{0}(x_{1}) \cdots \, d\mu_{0}(x_{r}) \\& \quad = \frac{(t+\log q)^{r}}{ (q^{a_{1}n}e^{a_{1} nt}-1 ) \cdots (q^{a_{r}n}e^{a_{r} nt}-1 )} \sum_{l_{1}, \ldots, l_{r}=0}^{n-1} q^{a_{1}l_{1}+\cdots+a_{r}l_{r}} e^{(a_{1}l_{1}+\cdots+a_{r}l_{r}) t} \\& \quad = \sum_{l_{1}, \ldots, l_{r}=0}^{n-1}\frac{(t+\log q)^{r}}{ (q^{a_{1}n}e^{a_{1} nt}-1 ) \cdots (q^{a_{r}n}e^{a_{r} nt}-1 )} q^{a_{1}l_{1}+\cdots+a_{r}l_{r}} e^{(a_{1}l_{1}+\cdots+a_{r}l_{r}) t} \\& \quad = \sum_{l_{1}, \ldots, l_{r}=0}^{n-1}q^{a_{1}l_{1}+\cdots+a_{r}l_{r}} \frac{(t+\log q)^{r}}{ (q^{a_{1}n}e^{a_{1} nt}-1 ) \cdots (q^{a_{r}n}e^{a_{r} nt}-1 )} e^{\frac{a_{1}l_{1}+\cdots+a_{r}l_{r}}{n} nt} \\& \quad = \sum_{l_{1}, \ldots, l_{r}=0}^{n-1}q^{a_{1}l_{1}+\cdots+a_{r}l_{r}} \sum_{m=0}^{\infty}B_{m,q^{n}} \biggl( \frac{a_{1}l_{1}+\cdots+a_{r}l_{r}}{n}\Big| a_{1},\ldots, a_{r} \biggr) \frac{n^{m}t^{m}}{m!} \\& \quad = \sum_{m=0}^{\infty}n^{m} \sum_{l_{1}, \ldots, l_{r}=0}^{n-1}q^{a_{1}l_{1}+\cdots+a_{r}l_{r}} B_{m,q^{n}} \biggl( \frac{a_{1}l_{1}+\cdots+a_{r}l_{r}}{n}\Big| a_{1},\ldots, a_{r} \biggr) \frac{t^{m}}{m!}. \end{aligned}$$
(19)

Thus, by (11) and (19), we obtain the following theorem.

Theorem 2.4

Let \(n\in\mathbb{N}\cup\{0\}\). Then we have
$$ B_{m,q}( a_{1}, \ldots, a_{r}) = n^{m} \sum_{l_{1}, \ldots, l_{r}=0}^{n-1}q^{a_{1}l_{1}+\cdots+a_{r}l_{r}} B_{m,q^{n}} \biggl( \frac{a_{1}l_{1}+\cdots+a_{r}l_{r}}{n}\Big| a_{1},\ldots, a_{r} \biggr) . $$
(20)

3 Higher order Barnes-type q-Euler polynomials

Higher Euler polynomials are defined as
$$ \biggl( \frac{2}{e^{t}+1} \biggr)^{r} e^{xt} =\sum_{n=0}^{\infty}E_{n} (x) \frac{t^{n}}{n!} \quad (\mbox{see [17--19, 22--24]}). $$
(21)
When \(x=0\), \(E_{n} =E_{n} (0)\) is called higher Euler numbers. For \(f\in \operatorname{UD}(\mathbb{Z}_{p})\), the fermionic p-adic integral on \(\mathbb {Z}_{p}\) is defined by Kim to be
$$ I_{-1}(f)= \int_{\mathbb{Z}_{p}} f(x) \, d \mu_{-1}(x) =\lim_{N\rightarrow\infty} \sum _{x=0}^{p^{N}-1} f(x) (-1)^{x} \quad (\mbox{see [4]}). $$
(22)
It is well known that an integral equation of the fermionic p-adic integral on \(\mathbb{Z}_{p}\) is
$$ I_{-1}(f_{1})+I_{-1}(f)=2f(0), $$
(23)
where \(f_{1}(x)=f(x+1)\).
Let \(a_{1}, \ldots, a_{r}\in\mathbb{C}_{p}\setminus\{0\}\). Higher order Barnes-type Euler polynomials are defined as
$$ \frac{2^{r}}{ (e^{a_{1}t}+1 )\cdots (e^{a_{r}t}+1 )} e^{xt} = \sum _{n=0}^{\infty}E_{n} (x|a_{1}, \ldots, a_{r}) \frac{t^{n}}{n!}\quad (\mbox{see [18, 19, 23]}). $$
(24)
When \(x=0\), \(E_{n}(a_{1}, \ldots, a_{r})=E_{n}(0|a_{1}, \ldots, a_{r})\) is called higher order Barnes-type Euler numbers. We define higher order Barnes-type q-Euler polynomials as follows:
$$ \frac{2^{r}}{ ( q^{a_{1}}e^{a_{1}t}+1 )\cdots ( q^{a_{r}}e^{a_{r}t}+1 )} e^{xt} = \sum _{n=0}^{\infty}E_{n,q}(x|a_{1}, \ldots, a_{r}) \frac{t^{n}}{n!}. $$
(25)
When \(x=0\), \(E_{n,q}(a_{1}, \ldots, a_{r})= E_{n,q}(0|a_{1}, \ldots, a_{r})\) is called higher order Barnes-type q-Euler numbers.
By (23), if we take \(f(x_{i})=q^{a_{i}x_{i}} e^{a_{i}x_{i}t}\) for \(i=1,\ldots,r\), then we have
$$ \int_{\mathbb{Z}_{p}} q^{a_{i}x_{i}} e^{a_{i}x_{i}t}\, d\mu_{-1}(x_{i})= \frac{2}{q^{a_{i}x_{i}}e^{a_{i}x_{i} t}+1}. $$
(26)
By (26), we get
$$\begin{aligned}& \int_{\mathbb{Z}_{p}} q^{a_{1}x_{1}+\cdots+a_{r}t_{r}} e^{(a_{1}x_{1}+\cdots +a_{r}x_{r} +x)t} \, d\mu_{0}(x_{1})\cdots\, d\mu_{0} (x_{r}) \\& \quad = \frac{2^{r}}{ ( q^{a_{1}}e^{a_{1}t}+1 )\cdots ( q^{a_{r}}e^{a_{r}t}+1 )} e^{xt}. \end{aligned}$$
(27)
By (24) and (27), we get
$$\begin{aligned}& \sum_{n=0}^{\infty}E_{n,q}(x|a_{1}, \ldots, a_{r})\frac{t^{n}}{n!} \\& \quad = \frac{2^{r}}{ ( q^{a_{1}}e^{a_{1}t}+1 )\cdots ( q^{a_{r}}e^{a_{r}t}+1 )} e^{xt} \\& \quad = \int_{\mathbb{Z}_{p}} \cdots\int_{\mathbb{Z}_{p}} q^{a_{1}x_{1}+\cdots +a_{r}x_{r}}e^{(a_{1}x_{1}+\cdots+a_{r}x_{r}+x)t} \, d\mu_{-1}(x_{1})\cdots\, d\mu_{-1}(x_{r}) \\& \quad = \sum_{n=0}^{\infty}\int _{\mathbb{Z}_{p}} \cdots\int_{\mathbb{Z}_{p}} q^{a_{1}x_{1}+\cdots+a_{r}x_{r}} (a_{1}x_{1}+\cdots+a_{r}x_{r}+x)^{n} \, d\mu_{-1}(x_{1})\cdots\, d\mu_{-1}(x_{r}) \frac{t^{n}}{n!}. \end{aligned}$$
(28)

From (28), we obtain the following theorem.

Theorem 3.1

Let \(n\in\mathbb{N}\cup\{0\}\). Then we have
$$\begin{aligned}& E_{n,q}(x| a_{1}, \ldots, a_{r}) \\& \quad = \int_{\mathbb{Z}_{p}} \cdots\int_{\mathbb{Z}_{p}} q^{a_{1}x_{1}+\cdots+a_{r}x_{r}} (a_{1}x_{1}+\cdots+a_{r}x_{r}+x)^{n} \, d\mu_{-1}(x_{1})\cdots\, d\mu_{-1}(x_{r}). \end{aligned}$$
(29)
From (22), we have
$$\begin{aligned} \begin{aligned}[b] \int_{\mathbb{Z}_{p}} f(x) \, d \mu_{-1}(x) &= \lim_{N\rightarrow\infty} \sum_{x=0}^{dp^{N}-1} f(x) (-1)^{x} \\ &= \frac{1}{d} \lim_{N\rightarrow\infty} \sum _{a=0}^{d-1} \sum_{x=0}^{p^{N}-1} (-1)^{a+x}f(a+dx) \\ &= \frac{1}{d}\sum_{a=0}^{d-1}(-1)^{a} \int_{\mathbb{Z}_{p}} f(a+dx)\, d \mu_{-1}(x). \end{aligned} \end{aligned}$$
(30)
By (30), if we take \(f(x_{i})=q^{a_{i}x_{i}}e^{a_{i}x_{i}t}\) for \(i=1, \ldots, r\), then we have
$$\begin{aligned} \int_{\mathbb{Z}_{p}} q^{a_{i}x_{i}}e^{a_{i}x_{i}t}\, d \mu_{-1}(x) =& \sum_{a=0}^{d-1}(-1)^{a} \int_{\mathbb{Z}_{p}} q^{a_{i}(a+dx-i)}e^{a_{i}(a+dx_{i})t} \, d \mu_{-1}(x_{i}) \\ =& \sum_{a=0}^{d-1}(-1)^{a} q^{a_{i} a}e^{a_{i} a t} \int_{\mathbb{Z}_{p}} q^{a_{i} dx_{i}}e^{a_{i} dx_{i}t}\, d \mu_{-1}(x_{i}). \end{aligned}$$
(31)
By (31), we get
$$\begin{aligned}& \int_{\mathbb{Z}_{p}} \cdots\int_{\mathbb{Z}_{p}} q^{a_{1}x_{1}+\cdots+a_{r}x_{r}} e^{(a_{1}x_{1}+\cdots+a_{r}x_{r}+x)t} \, d\mu_{-1}(x_{1})\cdots \, d\mu_{-1}(x_{r}) \\& \quad = \sum_{l_{1}, \ldots, l_{r}=0}^{d-1}(-1)^{l_{1}+\cdots +l_{r}}q^{a_{1}l_{1}+\cdots+a_{r}l_{r}} \\& \qquad {}\times\int_{\mathbb{Z}_{p}}\cdots\int_{\mathbb{Z}_{p}} q^{a_{1}\, dx_{1}+\cdots+a_{r}\, dx_{r}} e^{ (\frac{a_{1}l_{1}+\cdots+a_{1}l_{r} + x}{d}+ a_{1}x_{1}+\cdots+a_{r} x_{r} )\, dt} \, d\mu_{-1}(x_{1})\cdots \, d\mu_{-1}(x_{r}) \\& \quad = \sum_{n=0}^{\infty}d^{n} \sum_{l_{1}, \ldots, l_{r}=0}^{d-1}(-1)^{l_{1}+\cdots+l_{r}}q^{a_{1}l_{1}+\cdots+a_{r}l_{r}} \int_{\mathbb {Z}_{p}}\cdots\int_{\mathbb{Z}_{p}} q^{a_{1}\, dx_{1}+\cdots+a_{r}\, dx_{r}} \\& \qquad {}\times \biggl(\frac{a_{1}l_{1}+\cdots+a_{1}l_{r} + x}{d}+ a_{1}x_{1}+ \cdots+a_{r} x_{r} \biggr)^{n} \, d \mu_{-1}(x_{1})\cdots\, d\mu_{-1}(x_{r}) \frac{t^{n}}{n!} \\& \quad = \sum_{n=0}^{\infty}d^{n} \sum_{l_{1}, \ldots, l_{r}=0}^{d-1}(-1)^{l_{1}+\cdots+l_{r}}q^{a_{1}l_{1}+\cdots+a_{r}l_{r}} \\& \qquad {}\times E_{n,q^{d}} \biggl( \frac{a_{1}l_{1}+\cdots+a_{r}l_{r} + x}{d} \Big| a_{1}, \ldots, a_{r} \biggr) \frac{t^{n}}{n!}. \end{aligned}$$
(32)
By (27) and (32), we obtain the following theorem.

Theorem 3.2

Let \(n\in\mathbb{N}\cup\{0\}\). Then we have
$$\begin{aligned}& E_{n,q}(x| a_{1}, \ldots, a_{r}) \\& \quad = d^{n} \sum_{l_{1}, \ldots, l_{r}=0}^{d-1}(-1)^{l_{1}+\cdots +l_{r}}q^{a_{1}l_{1}+\cdots+a_{r}l_{r}} E_{n,q^{d}} \biggl( \frac{a_{1}l_{1}+\cdots +a_{r}l_{r} + x}{d} \Big| a_{1}, \ldots, a_{r} \biggr). \end{aligned}$$

Declarations

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Authors’ Affiliations

(1)
Graduate School of Education, Konkuk University
(2)
Department of Mathematics Education, Konkuk University
(3)
Department of Mathematics, Kwangwoon University

References

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© Jang et al. 2015