On the twisted Daehee polynomials with q-parameter

Abstract

The n th twisted Daehee numbers with q-parameter are closely related to higher-order Bernoulli numbers and Bernoulli numbers of the second kind. In this paper, we give a p-adic integral representation of the twisted Daehee polynomials with q-parameter, and we derive some interesting properties related to the n th twisted Daehee polynomials with q-parameter.

1 Introduction

Let p be a fixed prime number. Throughout this paper, ${\mathbb{Z}}_p$, ${\mathbb{Q}}_p$, and ${\mathbb{C}}_p$ will, respectively, denote the ring of p-adic rational integers, the field of p-adic rational numbers, and the completions of algebraic closure of ${\mathbb{Q}}_p$. The p-adic norm is defined ${|p|}_p=\frac{1}{p}$.

When one talks of q-extension, q is variously considered as an indeterminate, a complex $q\in \mathbb{C}$, or a p-adic number $q\in {\mathbb{C}}_p$. If $q\in \mathbb{C}$, one normally assumes that $|q|<1$. If $q\in {\mathbb{C}}_p$, then we assume that ${|q-1|}_p<p^{-\frac{1}{p-1}}$ so that $q^x=exp(xlogq)$ for each $x\in {\mathbb{Z}}_p$. Throughout this paper, we use the notation

$${[x]}_q=\frac{1-q^x}{1-q}.$$

Note that ${lim}_{q\to 1}{[x]}_q=x$ for each $x\in {\mathbb{Z}}_p$.

Let $UD({\mathbb{Z}}_p)$ be the space of uniformly differentiable functions on ${\mathbb{Z}}_p$. For $f\in UD({\mathbb{Z}}_p)$, the p-adic invariant integral on ${\mathbb{Z}}_p$ is defined by Kim as follows:

$$I(f)={\int }_{{\mathbb{Z}}_p}f(x)d{\mu }_0(x)=\underset{n\to \mathrm{\infty }}{lim}\frac{1}{p^n}\sum _{x=0}^{p^n-1}f(x)\text{(see [1–3])}.$$

(1.1)

Let $f_1$ be the translation of f with $f_1(x)=f(x+1)$. Then, by (1.1), we get

$$I(f_1)=I(f)+f^{\prime }(0),\text{where }{f}^{\prime }(0)=\frac{df(x)}{dx}{|}_{x=0}.$$

(1.2)

As is well known, the Stirling number of the first kind is defined by

$${(x)}_n=x(x-1)\cdots (x-n+1)=\sum _{l=0}^n{S}_1(n,l)x^l,$$

(1.3)

and the Stirling number of the second kind is given by the generating function to be

$${(e^t-1)}^m=m!\sum _{l=m}^{\mathrm{\infty }}{S}_2(l,m)\frac{t^l}{l!}$$

(1.4)

(see [4–6]).

Unsigned Stirling numbers of the first kind are given by

$$x^{\underline{n}}=x(x+1)\cdots (x+n-1)=\sum _{l=0}^n|S_1(n,l)|x^l.$$

(1.5)

Note that if we replace x to −x in (1.3), then

$$\begin{array}{rl}{(-x)}_n& ={(-1)}^n{x}^{\underline{n}}=\sum _{l=0}^n{S}_1(n,l){(-1)}^l{x}^l\\ ={(-1)}^n\sum _{l=0}^n|S_1(n,l)|x^l.\end{array}$$

(1.6)

Hence $S_1(n,l)=|S_1(n,l)|{(-1)}^{n-l}$.

For $r\in \mathbb{N}$, the Bernoulli polynomials of order r are defined by the generating function to be

$${\left(\frac{t}{e^t-1}\right)}^r{e}^{xt}=\sum _{n=0}^{\mathrm{\infty }}{B}_n^{(r)}(x)\frac{t^n}{n!}\text{(see [1, 4, 7–18])}.$$

(1.7)

When $x=0$, $B_n^{(r)}=B_n^{(r)}(0)$ are called the Bernoulli numbers of order r, and in the special case, $r=1$, $B_n^{(1)}(x)=B_n(x)$ are called the ordinary Bernoulli polynomials.

For $n\in \mathbb{N}$, let $T_p$ be the p-adic locally constant space defined by

$$T_p=\bigcup _{n\ge 1}{C}_{p^n}=\underset{n\to \mathrm{\infty }}{lim}{C}_{p^n},$$

where $C_{p^n}=\{\omega |{\omega }^{p^n}=1\}$ is the cyclic group of order $p^n$.

We assume that q is an indeterminate in ${\mathbb{C}}_p$ with ${|1-q|}_p<p^{-\frac{1}{p-1}}$. Then we define the q-analog of a falling factorial sequence as follows:

$${(x)}_{n,q}=x(x-q)(x-2q)\cdots (x-(n-1)q)(n\ge 1),{(x)}_{0,q}=1.$$

Note that

$$\underset{q\to 1}{lim}{(x)}_{n,q}={(x)}_n=\sum _{l=0}^n{S}_1(n,l)x^l.$$

Recently, DS Kim and T Kim introduced the Daehee polynomials as follows:

$$D_n(x)={\int }_{{\mathbb{Z}}_p}{(x+y)}_{n}d{\mu }_0(y)(n\ge 0)\text{(see [2, 9, 19])}.$$

(1.8)

When $x=0$, $D_n=D_n(0)$ are called the nth Daehee numbers. From (1.8), we can derive the generating function to be

$$\left(\frac{log(1+t)}{t}\right){(1+t)}^x=\sum _{n=0}^{\mathrm{\infty }}{D}_n(x)\frac{t^n}{n!}\text{(see [9])}.$$

(1.9)

In addition, DS Kim et al. consider the Daehee polynomials with q-parameter, which are defined by the generating function to be

$$\sum _{n=0}^{\mathrm{\infty }}{D}_{n,q}\frac{t^n}{n!}={(1+qt)}^{\frac{x}{q}}\frac{log(1+qt)}{q({(1+qt)}^{\frac{1}{q}}-1)}\text{(see [20, 21])}.$$

(1.10)

When $x=0$, $D_{n,q}=D_{n,q}(0)$ are called the Daehee numbers with q-parameter.

From the viewpoint of a generalization of the Daehee polynomials with q-parameter, we consider the twisted Daehee polynomials with q-parameter, defined to be

$$\sum _{n=0}^{\mathrm{\infty }}{D}_{n,\xi ,q}\frac{t^n}{n!}={(1+q\xi t)}^{\frac{x}{q}}\frac{log(1+q\xi t)}{q({(1+q\xi t)}^{\frac{1}{q}}-1)},$$

(1.11)

where $t,q\in {\mathbb{C}}_p$ with ${|t|}_p<{|q|}_p{p}^{-\frac{1}{p-1}}$ and $\xi \in T_p$.

In this paper, we give a p-adic integral representation of the twisted Daehee polynomials with q-parameter, which is called the Witt-type formula for the twisted Daehee polynomials with q-parameter. We can derive some interesting properties related to the n th twisted Daehee polynomials with q-parameter.

2 Witt-type formula for the n th twisted Daehee polynomials with q-parameter

First, we consider the following integral representation associated with falling factorial sequences:

$${\xi }^n{\int }_{{\mathbb{Z}}_p}{(x+y)}_{n,q}d{\mu }_0(y),\text{where }n\in {\mathbb{Z}}_{+}=\mathbb{N}\cup \{0\}\text{ and }\xi \in T_p.$$

(2.1)

By (2.1),

$$\begin{array}{rl}\sum _{n=0}^{\mathrm{\infty }}{\xi }^n{\int }_{{\mathbb{Z}}_p}{(x+y)}_{n,q}d{\mu }_0(y)\frac{t^n}{n!}& =\sum _{n=0}^{\mathrm{\infty }}{\xi }^n{q}^n{\int }_{{\mathbb{Z}}_p}{\left(\frac{x+y}{q}\right)}_{n}d{\mu }_0(y)\frac{t^n}{n!}\\ ={\int }_{{\mathbb{Z}}_p}{(1+q\xi t)}^{\frac{x+y}{q}}d{\mu }_0(y),\end{array}$$

(2.2)

where $t,q\in {\mathbb{C}}_p$ with ${|t|}_p<{|q|}_p{p}^{-\frac{1}{p-1}}$. For $t\in {\mathbb{C}}_p$ with ${|t|}_p<{|q|}_p{p}^{-\frac{1}{p-1}}$, put $f(x)={(1+q\xi t)}^{\frac{x+y}{q}}$. By (1.1), we get

$$\begin{array}{rl}{\int }_{{\mathbb{Z}}_p}{(1+q\xi t)}^{\frac{x+y}{q}}d{\mu }_0(y)& ={(1+q\xi t)}^{\frac{x}{q}}\frac{log(1+q\xi t)}{q({(1+q\xi t)}^{\frac{1}{q}}-1)}\\ =\sum _{n=0}^{\mathrm{\infty }}{D}_{n,\xi ,q}(x)\frac{t^n}{n!}.\end{array}$$

(2.3)

By (2.2) and (2.3), we obtain the following theorem.

Theorem 2.1 For $n\ge 0$, we have

$$D_{n,\xi ,q}(x)={\xi }^n{\int }_{{\mathbb{Z}}_p}{(x+y)}_{n,q}d{\mu }_0(y).$$

In (2.3), by replacing t by $\frac{1}{\xi q}(e^{\xi t}-1)$, we have

$$\sum _{n=0}^{\mathrm{\infty }}{D}_{n,\xi ,q}(x)\frac{1}{{\xi }^n{q}^n}\frac{{(e^{\xi t}-1)}^n}{n!}=e^{\frac{\xi tx}{q}}\frac{\frac{\xi t}{q}}{e^{\frac{\xi t}{q}}-1}=\sum _{n=0}^{\mathrm{\infty }}{B}_n(x)\frac{{\xi }^n}{q^n}\frac{t^n}{n!}$$

(2.4)

and

$$\begin{array}{rl}\sum _{n=0}^{\mathrm{\infty }}\frac{D_{n,\xi ,q}(x)}{{\xi }^n{q}^n}\frac{1}{n!}{(e^{\xi t}-1)}^n& =\sum _{n=0}^{\mathrm{\infty }}\frac{D_{n,\xi ,q}(x)}{{\xi }^n{q}^n}\sum _{m=n}^{\mathrm{\infty }}{\xi }^m{S}_2(m,n)\frac{t^m}{m!}\\ =\sum _{m=0}^{\mathrm{\infty }}\sum _{n=0}^m\frac{D_{n,\xi ,q}(x)}{{\xi }^n{q}^n}{\xi }^m{S}_2(m,n)\frac{t^m}{m!}.\end{array}$$

(2.5)

By (2.4) and (2.5), we obtain the following corollary.

Corollary 2.2 For $n\ge 0$, we have

$$B_n(x)=\sum _{m=0}^n{D}_{m,\xi ,q}(x){\xi }^{-m}{q}^{n-m}{S}_2(n,m).$$

By Theorem 2.1,

$$\begin{array}{rl}{D}_{n,\xi ,q}(x)& ={\xi }^n{\int }_{{\mathbb{Z}}_p}{(x+y)}_{n,q}d{\mu }_0(y)\\ ={\xi }^n{q}^n\sum _{l=0}^n\frac{1}{q^l}{S}_1(n,l){\int }_{{\mathbb{Z}}_p}{(x+y)}^{l}d{\mu }_0(y).\end{array}$$

(2.6)

By (1.2), we can derive easily that

$$\begin{array}{rl}{\int }_{{\mathbb{Z}}_p}{e}^{(x+y)t}d{\mu }_0(y)& =\frac{t}{e^t-1}{e}^{xt}=\sum _{n=0}^{\mathrm{\infty }}{B}_n(x)\frac{t^n}{n!}\\ =\sum _{l=0}^{\mathrm{\infty }}{\int }_{{\mathbb{Z}}_p}{(x+y)}^{l}d{\mu }_0(y)\frac{t^l}{l!},\end{array}$$

(2.7)

and so

$$B_n(x)={\int }_{{\mathbb{Z}}_p}{(x+y)}^{n}d{\mu }_0(y).$$

(2.8)

By (1.6), (2.7), and (2.8), we obtain the following corollary.

Corollary 2.3 For $n\ge 0$, we have

$$D_{n,\xi ,q}(x)={\xi }^n\sum _{l=0}^n{q}^{n-l}{S}_1(n,l)B_l(x)={\xi }^n\sum _{l=0}^n|S_1(n,l)|{(-q)}^{n-l}{B}_l(x).$$

From now on, we consider twisted Daehee polynomials of order $k\in \mathbb{N}$ with q-parameter. Twisted Daehee polynomials of order $k\in \mathbb{N}$ with q-parameter are defined by the multivariant p-adic invariant integral on ${\mathbb{Z}}_p$:

$$D_{n,\xi ,q}^{(k)}(x)={\xi }^n{\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}{(x_1+\cdots +x_k+x)}_{n,q}d{\mu }_0(x_1)\cdots d{\mu }_0(x_k),$$

(2.9)

where n is a nonnegative integer and $k\in \mathbb{N}$. In the special case, $x=0$, $D_{n,\xi ,q}^{(k)}=D_{n,\xi ,q}^{(k)}(0)$ are called the Daehee numbers of order k with q-parameter.

From (2.9), we can derive the generating function of $D_{n,\xi ,q}^{(k)}(x)$ as follows:

$$\begin{array}{r}\sum _{n=0}^{\mathrm{\infty }}{D}_{n,\xi ,q}^{(k)}(x)\frac{t^n}{n!}\\ =\sum _{n=0}^{\mathrm{\infty }}{\xi }^n{q}^n{\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}\left(\genfrac{}{}{0ex}{}{\frac{x_1+\cdots +x_k+x}{q}}{n}\right)d{\mu }_0(x_1)\cdots d{\mu }_0(x_k)t^n\\ ={\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}{(1+q\xi t)}^{\frac{x_1+\cdots +x_k+x}{q}}d{\mu }_0(x_1)\cdots d{\mu }_0(x_k)\\ ={(1+q\xi t)}^{\frac{x}{q}}{\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}{(1+q\xi t)}^{\frac{x_1+\cdots +x_k}{q}}d{\mu }_0(x_1)\cdots d{\mu }_0(x_k)\\ ={(1+q\xi t)}^{\frac{x}{q}}{\left(\frac{log(1+q\xi t)}{q({(1+q\xi t)}^{\frac{1}{q}}-1)}\right)}^k.\end{array}$$

(2.10)

Note that, by (2.9),

$$D_{n,\xi ,q}^{(k)}(x)={\xi }^n{q}^n\sum _{m=0}^n\frac{S_1(n,m)}{q^m}{\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}{(x_1+\cdots +x_k+x)}^{m}d{\mu }_0(x_1)\cdots d{\mu }_0(x_k).$$

(2.11)

Since

$${\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}{e}^{(x_1+\cdots +x_k+x)t}d{\mu }_0(x_1)\cdots d{\mu }_0(x_k)={\left(\frac{t}{e^t-1}\right)}^k{e}^{xt}=\sum _{n=0}^{\mathrm{\infty }}{B}_n^{(k)}(x)\frac{t^n}{n!},$$

we can derive easily

$$B_n^{(k)}(x)={\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}{(x_1+\cdots +x_k+x)}^{n}d{\mu }_0(x_1)\cdots d{\mu }_0(x_k).$$

(2.12)

Thus, by (2.11) and (2.12), we have

$$\begin{array}{rl}{D}_{n,\xi ,q}^{(k)}(x)& ={\xi }^n{q}^n\sum _{m=0}^n\frac{S_1(n,m)}{q^m}{B}_m^{(k)}(x)\\ ={\xi }^n\sum _{m=0}^n{q}^{n-m}{S}_1(n,m)B_m^{(k)}(x)\\ ={\xi }^n\sum _{m=0}^n|S_1(n,m)|{(-q)}^{n-m}{B}_m^{(k)}(x).\end{array}$$

(2.13)

In (2.10), by replacing t by $\frac{1}{q\xi }(e^{\xi t}-1)$, we get

$$\sum _{n=0}^{\mathrm{\infty }}{D}_{n,\xi ,q}^{(k)}(x)\frac{{(e^{\xi t}-1)}^n}{{\xi }^n{q}^{n}n!}=e^{\frac{\xi tx}{q}}{\left(\frac{\frac{\xi t}{q}}{e^{\frac{\xi t}{q}}-1}\right)}^k=\sum _{n=0}^{\mathrm{\infty }}\frac{{\xi }^n{B}_n^{(k)}(x)}{q^n}\frac{t^n}{n!}$$

(2.14)

and

$$\begin{array}{rl}\sum _{n=0}^{\mathrm{\infty }}\frac{D_{n,\xi ,q}^{(k)}(x)}{{\xi }^n{q}^n}\frac{1}{n!}{(e^{\xi t}-1)}^n& =\sum _{n=0}^{\mathrm{\infty }}\frac{D_{n,\xi ,q}^{(k)}(x)}{{\xi }^n{q}^n}\sum _{l=n}^{\mathrm{\infty }}{S}_2(l,n){\xi }^l\frac{t^l}{l!}\\ =\sum _{m=0}^{\mathrm{\infty }}({\xi }^m\sum _{n=0}^m\frac{D_{n,\xi ,q}^{(k)}(x)}{{\xi }^n{q}^n}{S}_2(m,n))\frac{t^m}{m!}.\end{array}$$

(2.15)

By (2.13), (2.14), and (2.15), we obtain the following theorem.

Theorem 2.4 For $n\ge 0$ and $k\in \mathbb{N}$, we have

$$D_{n,\xi ,q}^{(k)}(x)={\xi }^n\sum _{m=0}^n{q}^{n-m}{S}_1(n,m)B_m^{(k)}(x)={\xi }^n\sum _{m=0}^n|S_1(n,m)|{(-q)}^{n-m}{B}_m^{(k)}(x)$$

and

$$B_n^{(k)}(x)=\sum _{m=0}^n{D}_{m,\xi ,q}^{(k)}(x){\xi }^{-m}{q}^{n-m}{S}_2(n,m).$$

Now, we consider the twisted Daehee polynomials of the second kind with q-parameter as follows:

$${\hat{D}}_{n,\xi ,q}(x)={\xi }^n{\int }_{{\mathbb{Z}}_p}{(-y+x)}_{n,q}d{\mu }_0(y)(n\ge 0).$$

(2.16)

In the special case $x=0$, ${\hat{D}}_{n,\xi ,q}(0)={\hat{D}}_{n,\xi ,q}$ are called the twisted Daehee numbers of the second kind with q-parameter.

By (2.16), we have

$${\hat{D}}_{n,\xi ,q}(x)={\xi }^n{q}^n{\int }_{{\mathbb{Z}}_p}{\left(\frac{-y+x}{q}\right)}_{n}d{\mu }_0(y),$$

(2.17)

and so we can derive the generating function of ${\hat{D}}_{n,\xi ,q}(x)$ by (1.1) as follows:

$$\begin{array}{rl}\sum _{n=0}^{\mathrm{\infty }}{\hat{D}}_{n,\xi ,q}(x)\frac{t^n}{n!}=& \sum _{n=0}^{\mathrm{\infty }}{q}^n{\xi }^n{\int }_{{\mathbb{Z}}_p}{\left(\frac{-y+x}{q}\right)}_{n}d{\mu }_0(y)\frac{t^n}{n!}\\ =\sum _{n=0}^{\mathrm{\infty }}{q}^n{\xi }^n{\int }_{{\mathbb{Z}}_p}\left(\genfrac{}{}{0ex}{}{\frac{-y+x}{q}}{n}\right)d{\mu }_0(y)t^n\\ ={\int }_{{\mathbb{Z}}_p}{(1+q\xi t)}^{\frac{-y+x}{q}}d{\mu }_0(y)\\ ={(1+q\xi t)}^{\frac{x}{q}}\frac{log(1+q\xi t)}{q({(1+q\xi t)}^{\frac{1}{q}}-1)}{(1+q\xi t)}^{\frac{1}{q}}.\end{array}$$

(2.18)

From (1.3), (1.6), and (2.17), we get

$$\begin{array}{rl}{\hat{D}}_{n,\xi ,q}(x)& =q^n{\xi }^n{\int }_{{\mathbb{Z}}_p}{\left(\frac{-y+x}{q}\right)}_{n}d{\mu }_0(y)\\ =q^n{\xi }^n{\int }_{{\mathbb{Z}}_p}\sum _{l=0}^n\frac{S_1(n,l)}{q^l}{(-y+x)}^{l}d{\mu }_0(y)\\ ={\xi }^n\sum _{l=0}^n{S}_1(n,l){(-1)}^l{\int }_{{\mathbb{Z}}_p}{(y-x)}^{l}d{\mu }_0(y)q^{n-l}\\ ={\xi }^n\sum _{l=0}^n{S}_1(n,l){(-1)}^l{B}_l(-x)q^{n-l}\\ ={(-\xi )}^n\sum _{l=0}^n|S_1(n,l)|B_l(-x)q^{n-l}.\end{array}$$

(2.19)

By (1.10), it is easy to show that $B_n(-x)={(-1)}^n{B}_n(x+1)$. Thus, from (2.19), we have the following theorem.

Theorem 2.5 For $n\ge 0$, we have

$${\hat{D}}_{n,\xi ,q}(x)={\xi }^n\sum _{l=0}^n{S}_1(n,l){(-1)}^l{B}_l(-x)q^{n-l}={\xi }^n\sum _{l=0}^n|S_1(n,l)|B_l(x+1){(-q)}^{n-l}.$$

By replacing t by $\frac{1}{q\xi }(e^{\xi t}-1)$ in (2.18), we have

$$\sum _{n=0}^{\mathrm{\infty }}{\hat{D}}_{n,\xi ,q}(x)\frac{1}{q^n{\xi }^n}\frac{{(e^{\xi t}-1)}^n}{n!}=e^{\frac{\xi t}{q}(x+1)}\frac{\frac{\xi t}{q}}{e^{\frac{\xi t}{q}}-1}=\sum _{n=0}^{\mathrm{\infty }}\frac{{\xi }^n{B}_n(x+1)}{q^n}\frac{t^n}{n!}$$

(2.20)

and

$$\begin{array}{rl}\sum _{n=0}^{\mathrm{\infty }}{\hat{D}}_{n,\xi ,q}(x)\frac{1}{q^n{\xi }^n}\frac{{(e^{\xi t}-1)}^n}{n!}& =\sum _{n=0}^{\mathrm{\infty }}\frac{{\hat{D}}_{n,\xi ,q}(x)}{q^n{\xi }^n}\sum _{m=n}^{\mathrm{\infty }}{S}_2(m,n)\frac{{(\xi t)}^m}{m!}\\ =\sum _{n=0}^{\mathrm{\infty }}(\sum _{m=0}^n{\hat{D}}_{m,\xi ,q}(x)S_2(n,m)q^{-m}{\xi }^{n-m})\frac{t^n}{n!}.\end{array}$$

(2.21)

By (2.20) and (2.21), we obtain the following theorem.

Theorem 2.6 For $n\ge 0$, we have

$$B_n(x+1)=\sum _{m=0}^n{q}^{n-m}{\xi }^{-m}{\hat{D}}_{m,\xi ,q}(x)S_2(n,m).$$

Now, we consider higher-order twisted Daehee polynomials of the second kind with q-parameter. Higher-order twisted Daehee polynomials of the second kind with q-parameter are defined by the multivariant p-adic invariant integral on ${\mathbb{Z}}_p$:

$${\hat{D}}_{n,\xi ,q}^{(k)}(x)={\xi }^n{\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}{(-x_1-\cdots -x_k+x)}_{n,q}d{\mu }_0(x_1)\cdots d{\mu }_0(x_k),$$

(2.22)

where n is a nonnegative integer and $k\in \mathbb{N}$. In the special case, $x=0$, ${\hat{D}}_{n,\xi ,q}^{(k)}={\hat{D}}_{n,\xi ,q}^{(k)}(0)$ are called the higher-order twisted Daehee numbers of the second kind with q-parameter.

From (2.22), we can derive the generating function of ${\hat{D}}_{n,\xi ,q}^{(k)}(x)$ as follows:

$$\begin{array}{r}\sum _{n=0}^{\mathrm{\infty }}{\hat{D}}_{n,\xi ,q}^{(k)}(x)\frac{t^n}{n!}\\ =\sum _{n=0}^{\mathrm{\infty }}{\xi }^n{q}^n{\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}\left(\genfrac{}{}{0ex}{}{\frac{-x_1-\cdots -x_k+x}{q}}{n}\right)d{\mu }_0(x_1)\cdots d{\mu }_0(x_k)t^n\\ ={\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}{(1+q\xi t)}^{\frac{-x_1-\cdots -x_k+x}{q}}d{\mu }_0(x_1)\cdots d{\mu }_0(x_k)\\ ={(1+q\xi t)}^{\frac{x+k}{q}}{\left(\frac{log(1+q\xi t)}{q({(1+q\xi t)}^{\frac{1}{q}}-1)}\right)}^k.\end{array}$$

(2.23)

By (2.22),

$$\begin{array}{r}{\hat{D}}_{n,\xi ,q}^{(k)}(x)\\ ={\xi }^n{q}^n\sum _{m=0}^n\frac{S_1(n,m)}{q^m}{\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}{(-x_1-\cdots -x_k+x)}^{m}d{\mu }_0(x_1)\cdots d{\mu }_0(x_k)\\ ={\xi }^n{q}^n\sum _{m=0}^n\frac{S_1(n,m)}{{(-q)}^m}{\int }_{{\mathbb{Z}}_p}\cdots {\int }_{{\mathbb{Z}}_p}{(x_1+\cdots +x_k-x)}^{m}d{\mu }_0(x_1)\cdots d{\mu }_0(x_k)\\ ={\xi }^n{q}^n\sum _{m=0}^n\frac{S_1(n,m)}{{(-q)}^m}{B}_m^{(k)}(-x)\\ ={\xi }^n\sum _{m=0}^n{q}^{n-m}|S_1(n,m)|B_m^{(k)}(-x).\end{array}$$

(2.24)

From (1.10), we know that $B_n^{(k)}(-x)={(-1)}^n{B}_n^{(k)}(k+x)$. Hence, by (2.24), we obtain the following theorem.

Theorem 2.7 For $n\ge 0$, we have

$${\hat{D}}_{n,\xi ,q}^{(k)}(x)={\xi }^n\sum _{m=0}^n{(-1)}^m{q}^{n-m}{S}_1(n,m)B_m^{(k)}(-x)={\xi }^n\sum _{m=0}^n{(-1)}^m{q}^{n-m}|S_1(n,m)|B_m^{(k)}(x+k).$$

In (2.23), by replacing t by $\frac{1}{q\xi }(e^{\xi t}-1)$, we get

$$\sum _{n=0}^{\mathrm{\infty }}{\hat{D}}_{n,\xi ,q}^{(k)}(x)\frac{{(e^{\xi t}-1)}^n}{{\xi }^n{q}^{n}n!}=e^{\frac{\xi t}{q}(x+k)}{\left(\frac{\frac{\xi t}{q}}{e^{\frac{\xi t}{q}}-1}\right)}^k=\sum _{n=0}^{\mathrm{\infty }}\frac{{\xi }^n{B}_n^{(k)}(x+k)}{q^n}\frac{t^n}{n!}$$

(2.25)

and

$$\begin{array}{rl}\sum _{n=0}^{\mathrm{\infty }}\frac{{\hat{D}}_{n,\xi ,q}^{(k)}(x)}{{\xi }^n{q}^n}\frac{1}{n!}{(e^{\xi t}-1)}^n& =\sum _{n=0}^{\mathrm{\infty }}\frac{{\hat{D}}_{n,\xi ,q}^{(k)}(x)}{{\xi }^n{q}^n}\sum _{l=n}^{\mathrm{\infty }}{S}_2(l,n){\xi }^l\frac{t^l}{l!}\\ =\sum _{n=0}^{\mathrm{\infty }}({\xi }^n\sum _{m=0}^n\frac{{\hat{D}}_{m,\xi ,q}^{(k)}(x)}{{\xi }^m{q}^m}{S}_2(n,m))\frac{t^n}{n!}.\end{array}$$

(2.26)

By (2.25) and (2.26), we obtain the following theorem.

Theorem 2.8 For $n\ge 0$ and $k\in \mathbb{N}$, we have

$$B_n^{(k)}(x+k)=\sum _{m=0}^n{\hat{D}}_{m,\xi ,q}^{(k)}(x){\xi }^{-m}{q}^{n-m}{S}_2(n,m).$$