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

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.

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

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