Some identities on the weighted q-Euler numbers and q-Bernstein polynomials

Abstract

Recently, Ryoo introduced the weighted q-Euler numbers and polynomials which are a slightly different Kim's weighted q-Euler numbers and polynomials(see C. S. Ryoo, A note on the weighted q-Euler numbers and polynomials, 2011]). In this paper, we give some interesting new identities on the weighted q-Euler numbers related to the q-Bernstein polynomials

2000 Mathematics Subject Classification - 11B68, 11S40, 11S80

1. Introduction

Let p be a fixed odd prime number. Throughout this paper ℤ _p , ${\mathbb{Q}}_p$, ℂ and ℂ _p will denote the ring of p-adic integers, the field of p-adic rational numbers, the complex number fields and the completion of algebraic closure of ${\mathbb{Q}}_p$, respectively. Let ℕ be the set of natural numbers and ℤ₊ = ℕ ∪ {0}. Let ν _p be the normalized exponential valuation of ℂ _p with $|p{|}_p=p^{-{\nu }_p\left(p\right)}=\frac{1}{p}$. When one talks of q-extension, q is variously considered as an indeterminate, a complex number q ∈ ℂ, or a p-adic number q ∈ ℂ _p . If q ∈ ℂ, then one normally assumes |q| < 1, and if q ∈ ℂ _p , then one normally assumes |q - 1| _p < 1. In this paper, the q-number is defined by

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

(see [1–19])

Note that lim_q→1[x] _q = x (see [1–19]). Let f be a continuous function on ℤ _p . For α ∈ ℕ and k, n ∈ ℤ₊, the weighted p-adic q-Bernstein operator of order n for f is defined by Kim as follows:

$$\begin{array}{lll}\hfill B_{n,q}^{\left(\alpha \right)}\left(f|x\right)& =\sum _{k=0}^n\left(\begin{array}{c}\hfill n\hfill \\ \hfill k\hfill \end{array}\right)f\left(\frac{k}{n}\right){\left[x\right]}_{q^{\alpha }}^k{\left[1-x\right]}_{q^{-\alpha }}^{n-k}& \hfill \text{(1)}\\ =\sum _{k=0}^{n}f\left(\frac{k}{n}\right)B_{k,n}^{\left(\alpha \right)}\left(x,q\right),.& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}$$

(1)

see [4, 9, 19].

Here $B_{k,n}^{\left(\alpha \right)}\left(x,q\right)=\left(\begin{array}{c}\hfill n\hfill \\ \hfill k\hfill \end{array}\right){\left[x\right]}_{q^{\alpha }}^k{\left[1-x\right]}_{q^{-\alpha }}^{n-k}$ are called the q-Bernstein polynomials of degree n with weighted α.

Let C(ℤ _p ) be the space of continuous functions on ℤ _p . For f ∈ C(ℤ _p ), the fermionic q-integral on ℤ _p is defined by

$$I_q(f)={\displaystyle {\int }_{{\mathbb{Z}}_p}}f(x)d{\mu }_{-q}(x)=\underset{N\to \infty }{{\displaystyle lim}}\frac{1+q}{1+q^{p^N}}{\displaystyle \sum _{x=0}^{p^N-1}}f(x)(-q{)}^x\mathrm{,}$$

(2)

see [5–19].

For n ∈ ℕ, by (2), we get

$$q^n{\displaystyle {\int }_{{\mathbb{Z}}_p}}f(x+n)d{\mu }_{-q}(x)=(-{1)}^n{\displaystyle {\int }_{{\mathbb{Z}}_p}}f(x)d{\mu }_{-q}(x)+{[2]}_q{\displaystyle \sum _{l=0}^{n-1}}{(-1)}^{n-1-l}{q}^{l}f(l)\mathrm{,}$$

(3)

see [6, 7].

Recently, by (2) and (3), Ryoo considered the weighted q-Euler polynomials which are a slightly different Kim's weighted q-Euler polynomials as follows:

$${\int }_{{\mathbb{Z}}_p}{\left[x+y\right]}_{q^{\alpha }}^{n}d{\mu }_{-q}\left(y\right)=E_{n,q}^{\left(\alpha \right)}\left(x\right),\mathsf{\text{for}}n\in {\mathbb{Z}}_{+}\mathsf{\text{and}}\alpha \in \mathbb{Z},$$

(4)

see [17].

In the special case, x = 0, $E_{n,q}^{\left(\alpha \right)}\left(0\right)=E_{n,q}^{\left(\alpha \right)}$ are called the n-th q-Euler numbers with weight α (see [14]).

From (4), we note that

$$E_{n,q}^{\left(\alpha \right)}\left(x\right)=\frac{{\left[2\right]}_q}{{\left(1-q^{\alpha }\right)}^n}\sum _{l=0}^n\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^l\frac{q^{\alpha lx}}{1+q^{\alpha l+1}},$$

(5)

see [17].

and

$$E_{n,q}^{\left(\alpha \right)}\left(x\right)=\sum _{l=0}^n\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right){\left[x\right]}_{q^{\alpha }}^{n-l}{q}^{\alpha lx}{E}_{l,q}^{\left(\alpha \right)},$$

(6)

see [17].

That is, (6) can be written as

$$E_{n\mathrm{,}q}^{(\alpha )}(x)=(q^{\alpha x}{E}_q^{(\alpha )}+{[x]}_{q^{\alpha }}{)}^n\mathrm{,}n\in {\mathbb{Z}}_{+}\mathrm{.}$$

(7)

with usual convention about replacing ${\left(E_q^{\left(\alpha \right)}\right)}^n$ by $E_{n,q}^{\left(\alpha \right)}$.

In this paper we study the weighted q-Bernstein polynomials to express the fermionic q-integral on ℤ _p and investigate some new identities on the weighted q-Euler numbers related to the weighted q-Bernstein polynomials.

2. q-Euler numbers with weight α

In this section we assume that α ∈ ℕ and q ∈ ℂ with |q| < 1.

Let F _q (t, x) be the generating function of q-Euler polynomials with weight α as followings:

$$F_q\left(t,x\right)=\sum _{n=0}^{\infty }{E}_{n,q}^{\left(\alpha \right)}\left(x\right)\frac{t^n}{n!}.$$

(8)

By (5) and (8), we get

$$\begin{array}{lll}\hfill F_q\left(t,x\right)& =\sum _{n=0}^{\infty }\left(\frac{{\left[2\right]}_q}{{\left(1-q^{\alpha }\right)}^n}\sum _{l=0}^n\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^l\frac{q^{\alpha lx}}{1+q^{\alpha l+1}}\right)\frac{t^n}{n!}& \hfill \text{(1)}\\ ={\left[2\right]}_q\sum _{m=0}^{\infty }{\left(-1\right)}^m{q}^m{e}^{{\left[x+m\right]}_q\alpha t}.& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}$$

(9)

In the special case, x = 0, let F _q (t, 0) = F _q (t). Then we obtain the following difference equation.

$$q{F}_q\left(t,1\right)+F_q\left(t\right)={\left[2\right]}_q.$$

(10)

Therefore, by (8) and (10), we obtain the following proposition.

Proposition 1. For n ∈ ℤ₊, we have

$$E_{0,q}^{\left(\alpha \right)}=1,\mathsf{\text{and}}q{E}_{n,q}^{\left(\alpha \right)}\left(1\right)+E_{n,q}^{\left(\alpha \right)}=0\mathsf{\text{if}}n>0.$$

By (6), we easily get the following corollary.

Corollary 2. For n ∈ ℤ₊, we have

$$E_{0,q}^{\left(\alpha \right)}=1,\mathsf{\text{and}}q{\left(q^{\alpha }{E}_q^{\left(\alpha \right)}+1\right)}^n+E_{n,q}^{\left(\alpha \right)}=0\mathsf{\text{if}}n>0,$$

with usual convention about replacing ${\left(E_q^{\left(\alpha \right)}\right)}^n$ by $E_{n,q}^{\left(\alpha \right)}$.

From (9), we note that

$$F_{q^{-1}}\left(t,1-x\right)=F_q\left(-q^{\alpha }t,x\right).$$

(11)

Therefore, by (11), we obtain the following lemma.

Lemma 3. Let n ∈ ℤ₊. Then we have

$$E_{n,q^{-1}}^{\left(\alpha \right)}\left(1-x\right)=(-1{)}^n{q}^{\alpha n}{E}_{n,q}^{\left(\alpha \right)}\left(x\right).$$

By Corollary 2, we get

$$\begin{array}{lll}\hfill q^2{E}_{n,q}^{\left(\alpha \right)}\left(2\right)-q^2-q& =q^2\sum _{l=0}^n\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right)q^{\alpha l}{\left(q^{\alpha }{E}_q^{\left(\alpha \right)}+1\right)}^l-q^2-q& \hfill \text{(1)}\\ =-q\sum _{l=1}^n\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right)q^{\alpha l}{E}_{l,q}^{\left(\alpha \right)}-q& \hfill \text{(2)}\\ =-q\sum _{l=0}^n\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right)q^{\alpha l}{E}_{l,q}^{\left(\alpha \right)}& \hfill \text{(3)}\\ =-q{E}_{n,q}^{\left(\alpha \right)}\left(1\right)=E_{n,q}^{\left(\alpha \right)}\mathsf{\text{if}}n>0.& \hfill \text{(4)}\\ \hfill \text{(5)}\end{array}$$

(12)

Therefore, by (12), we obtain the following theorem.

Theorem 4. For n ∈ ℕ, we have

$$E_{n,q}^{\left(\alpha \right)}\left(2\right)=\frac{1}{q^2}{E}_{n,q}^{\left(\alpha \right)}+\frac{1}{q}+1.$$

Theorem 4 is important to study the relations between q-Bernstein polynomials and the weighted q-Euler number in the next section.

3. Weighted q-Euler numbers concerning q-Bernstein polynomials

In this section we assume that α ∈ ℤ _p and q ∈ ℂ _p with |1 - q| _p < 1.

From (2), (3) and (4), we note that

$$\begin{array}{c}q{\displaystyle {\int }_{{\mathbb{Z}}_p}}{[1-x]}_{q^{-\alpha }}^{n}d{\mu }_{-q}(x)=(-{1)}^n{q}^{\alpha n+1}{\displaystyle {\int }_{{\mathbb{Z}}_p}}{[x-1]}_{q^{\alpha }}^{n}d{\mu }_{-q}(x)\\ =q{\displaystyle \sum _{l=0}^n}\left(\begin{array}{c}n\\ l\end{array}\right){(-1)}^l{\displaystyle {\int }_{{\mathbb{Z}}_p}}{[x]}_{q^{\alpha }}^{l}d{\mu }_{-q}(x)\mathrm{.}\end{array}$$

(13)

Therefore, by (13) and Lemma 3, we obtain the following theorem.

Theorem 5. For n ∈ ℤ₊, we get

$$\begin{array}{c}q{\displaystyle {\int }_{{\mathbb{Z}}_p}}{[1-x]}_{q^{-\alpha }}^{n}d{\mu }_{-q}(x)=(-{1)}^n{q}^{\alpha n+1}{E}_{n\mathrm{,}q}^{(\alpha )}(-1)=q{E}_{n\mathrm{,}{q}^{-1}}^{(\alpha )}(2)\\ =q{\displaystyle \sum _{l=0}^n}\left(\begin{array}{c}n\\ l\end{array}\right){(-1)}^l{E}_{l\mathrm{,}q}^{(\alpha )}\mathrm{.}\end{array}$$

Let n ∈ ℕ. Then, by Theorem 4, we obtain the following corollary.

Corollary 6. For n ∈ ℕ, we have

$$\begin{array}{lll}\hfill {\int }_{{\mathbb{Z}}_p}{\left[1-x\right]}_{q^{-\alpha }}^{n}d{\mu }_{-q}\left(x\right)& =E_{n,q^{-1}}^{\left(\alpha \right)}\left(2\right)& \hfill \text{(1)}\\ =q^2{E}_{n,q^{-1}}^{\left(\alpha \right)}+{\left[2\right]}_q.& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}$$

For x ∈ ℤ _p , the p-adic q-Bernstein polynomials with weight α of degree n are given by

$$B_{k,n}^{\left(\alpha \right)}\left(x,q\right)=\left(\begin{array}{c}\hfill n\hfill \\ \hfill k\hfill \end{array}\right){\left[x\right]}_{q^{\alpha }}^k{\left[1-x\right]}_{q^{-\alpha }}^{n-k},\mathsf{\text{where}}n,k\in {\mathbb{Z}}_{+},$$

(14)

see [9].

From (14), we can easily derive the following symmetric property for q-Bernstein polynomials:

$$B_{k,n}^{\left(\alpha \right)}\left(x,q\right)=B_{n-k,n}^{\left(\alpha \right)}\left(1-x,q^{-1}\right),$$

(15)

see [11]

By (15), we get

$$\begin{array}{lll}\hfill {\int }_{{\mathbb{Z}}_p}{B}_{k,n}^{\left(\alpha \right)}\left(x,q\right)d{\mu }_{-q}\left(x\right)& ={\int }_{{\mathbb{Z}}_p}{B}_{n-k,n}^{\left(\alpha \right)}\left(1-x,q^{-1}\right)d{\mu }_{-q}\left(x\right)& \hfill \text{(1)}\\ =\left(\begin{array}{c}\hfill n\hfill \\ \hfill k\hfill \end{array}\right)\sum _{l=0}^k\left(\begin{array}{c}\hfill k\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{k+l}{\int }_{{\mathbb{Z}}_p}{\left[1-x\right]}_{q^{-\alpha }}^{n-l}d{\mu }_{-q}\left(x\right).& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}$$

(16)

Let n, k ∈ ℤ₊ with n > k. Then, by (16) and Corollary 6, we have

$$\begin{array}{c}{\int }_{{\mathbb{Z}}_p}{B}_{k,n}^{\left(\alpha \right)}\left(x,q\right)d{\mu }_{-q}\left(x\right)\\ =\left(\begin{array}{c}\hfill n\hfill \\ \hfill k\hfill \end{array}\right)\sum _{l=0}^k\left(\begin{array}{c}\hfill k\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{k+l}\left(q^2{E}_{n-l,q^{-1}}^{\left(\alpha \right)}+{\left[2\right]}_q\right)\\ =\left\{\begin{array}{cc}\hfill q^2{E}_{n,q^{-1}}^{\left(\alpha \right)}+{\left[2\right]}_q,\hfill & \hfill \mathsf{\text{if }}k=0,\hfill \\ \hfill q^2\left(\begin{array}{c}\hfill n\hfill \\ \hfill k\hfill \end{array}\right){\sum }_{l=0}^k\left(\begin{array}{c}\hfill k\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{k+l}{E}_{n-l,q^{-1}}^{\left(\alpha \right)},\hfill & \hfill \mathsf{\text{if }}k>0.\hfill \end{array}\right.\end{array}$$

(17)

Taking the fermionic q-integral on ℤ _p for one weighted q-Bernstein polynomials in (14), we have

$$\begin{array}{lll}\hfill {\int }_{{\mathbb{Z}}_p}{B}_{k,n}^{\left(\alpha \right)}\left(x,q\right)d{\mu }_{-q}\left(x\right)& =\left(\begin{array}{c}\hfill n\hfill \\ \hfill k\hfill \end{array}\right){\int }_{{\mathbb{Z}}_p}{\left[x\right]}_{q^{\alpha }}^k{\left[1-x\right]}_{q^{-\alpha }}^{n-k}d{\mu }_{-q}\left(x\right)& \hfill \text{(1)}\\ =\left(\begin{array}{c}\hfill n\hfill \\ \hfill k\hfill \end{array}\right)\sum _{l=0}^{n-k}\left(\begin{array}{c}\hfill n-k\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^l{\int }_{{\mathbb{Z}}_p}{\left[x\right]}_{q^{\alpha }}^{k+l}d{\mu }_{-q}\left(x\right)& \hfill \text{(2)}\\ =\left(\begin{array}{c}\hfill n\hfill \\ \hfill k\hfill \end{array}\right)\sum _{l=0}^{n-k}\left(\begin{array}{c}\hfill n-k\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^l{E}_{l+k,q}^{\left(\alpha \right)}.& \hfill \text{(3)}\\ \hfill \text{(4)}\end{array}$$

(18)

Therefore, by comparing the coefficients on the both sides of (17) and (18), we obtain the following theorem.

Theorem 7. For n, k ∈ ℤ₊ with n > k, we have

$$\sum _{l=0}^{n-k}{\left(-1\right)}^l\left(\begin{array}{c}\hfill n-k\hfill \\ \hfill l\hfill \end{array}\right)E_{l+k,q}^{\left(\alpha \right)}=\left\{\begin{array}{cc}\hfill q^2{E}_{n,q^{-1}}^{\left(\alpha \right)}+{\left[2\right]}_q,\hfill & \hfill \mathsf{\text{if}}k=0,\hfill \\ \hfill q^2{\sum }_{l=0}^k\left(\begin{array}{c}\hfill k\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{k+l}{E}_{n-l,q^{-1}}^{\left(\alpha \right)},\hfill & \hfill \mathsf{\text{if }}k>0.\hfill \end{array}\right.$$

Let n₁, n₂, k ∈ ℤ₊ with n₁ + n₂ > 2k. Then we see that

$$\begin{array}{c}{\int }_{{\mathbb{Z}}_p}{B}_{k,n_1}^{\left(\alpha \right)}\left(x,q\right)B_{k,n_2}^{\left(\alpha \right)}\left(x,q\right)d{\mu }_{-q}\left(x\right)\\ =\left(\begin{array}{c}\hfill n_1\hfill \\ \hfill k\hfill \end{array}\right)\left(\begin{array}{c}\hfill n_2\hfill \\ \hfill k\hfill \end{array}\right)\sum _{l=0}^{2k}\left(\begin{array}{c}\hfill 2k\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{l+2k}{\int }_{{\mathbb{Z}}_p}{\left[1-x\right]}_{q^{-\alpha }}^{n_1+n_2-l}d{\mu }_{-q}\left(x\right)\\ =\left(\begin{array}{c}\hfill n_1\hfill \\ \hfill k\hfill \end{array}\right)\left(\begin{array}{c}\hfill n_2\hfill \\ \hfill k\hfill \end{array}\right)\sum _{l=0}^{2k}\left(\begin{array}{c}\hfill 2k\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{l+2k}\left(q^2{E}_{n_1+n_2-l,q^{-1}}^{\left(\alpha \right)}+{\left[2\right]}_q\right).\end{array}$$

(19)

By the binomial theorem and definition of q-Bernstein polynomials, we get

$$\begin{array}{c}{\int }_{{\mathbb{Z}}_p}{B}_{k,n_1}^{\left(\alpha \right)}\left(x,q\right)B_{k,n_2}^{\left(\alpha \right)}\left(x,q\right)d{\mu }_{-q}\left(x\right)\\ =\left(\begin{array}{c}\hfill n_1\hfill \\ \hfill k\hfill \end{array}\right)\left(\begin{array}{c}\hfill n_2\hfill \\ \hfill k\hfill \end{array}\right)\sum _{l=0}^{n_1+n_2-2k}{\left(-1\right)}^l\left(\begin{array}{c}\hfill n_1+n_2-2k\hfill \\ \hfill l\hfill \end{array}\right){\int }_{{\mathbb{Z}}_p}{\left[x\right]}_{q^{\alpha }}^{2k+l}d{\mu }_{-q}\left(x\right)\\ =\left(\begin{array}{c}\hfill n_1\hfill \\ \hfill k\hfill \end{array}\right)\left(\begin{array}{c}\hfill n_2\hfill \\ \hfill k\hfill \end{array}\right)\sum _{l=0}^{n_1+n_2-2k}{\left(-1\right)}^l\left(\begin{array}{c}\hfill n_1+n_2-2k\hfill \\ \hfill l\hfill \end{array}\right)E_{2k+l,q}^{\left(\alpha \right)}.\end{array}$$

(20)

By comparing the coefficients on the both sides of (19) and (20), we obtain the following theorem.

Theorem 8. Let n₁, n₂, k ∈ ℤ₊ with n₁ + n₂ > 2k. Then we have

$$\begin{array}{c}\sum _{l=0}^{n_1+n_2-2k}{\left(-1\right)}^l\left(\begin{array}{c}\hfill n_1+n_2-2k\hfill \\ \hfill l\hfill \end{array}\right)E_{2k+l,q}^{\left(\alpha \right)}\\ =\left\{\begin{array}{cc}\hfill q^2{E}_{n_1+n_2,q^{-1}}^{\left(\alpha \right)}+{\left[2\right]}_q,\hfill & \hfill \mathsf{\text{if }}k=0,\hfill \\ \hfill q^2{\sum }_{l=0}^{2k}\left(\begin{array}{c}\hfill 2k\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{2k+l}{E}_{n_1+n_2-l,q^{-1}}^{\left(\alpha \right)},\hfill & \hfill \mathsf{\text{if }}k>0.\hfill \end{array}\right.\end{array}$$

Let s ∈ ℕ with s ≥ 2. For n₁, n₂, ..., n _s , k ∈ ℤ₊ with n₁ + ⋯ + n _s > sk, we have

$$\begin{array}{c}{\int }_{{\mathbb{Z}}_p}\underset{s-times}{\underbrace{B_{k,n_1}^{\left(\alpha \right)}\left(x,q\right)\cdots B_{k,n_s}^{\left(\alpha \right)}\left(x,q\right)}}d{\mu }_{-q}\left(x\right)\\ =\left(\begin{array}{c}\hfill n_1\hfill \\ \hfill k\hfill \end{array}\right)\cdots \left(\begin{array}{c}\hfill n_s\hfill \\ \hfill k\hfill \end{array}\right){\int }_{{\mathbb{Z}}_p}{\left[x\right]}_{q^{\alpha }}^{sk}{\left[1-x\right]}_{q^{-\alpha }}^{n_1+\cdots +n_s-sk}d{\mu }_{-q}\left(x\right)\\ =\left(\begin{array}{c}\hfill n_1\hfill \\ \hfill k\hfill \end{array}\right)\cdots \left(\begin{array}{c}\hfill n_s\hfill \\ \hfill k\hfill \end{array}\right)\sum _{l=0}^{sk}\left(\begin{array}{c}\hfill sk\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{l+sk}{\int }_{{\mathbb{Z}}_p}{\left[1-x\right]}_{q^{-\alpha }}^{n_1+\cdots +n_s-l}d{\mu }_{-q}\left(x\right)\\ =\left(\begin{array}{c}\hfill n_1\hfill \\ \hfill k\hfill \end{array}\right)\cdots \left(\begin{array}{c}\hfill n_s\hfill \\ \hfill k\hfill \end{array}\right)\sum _{l=0}^{sk}\left(\begin{array}{c}\hfill sk\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{l+sk}\left(q^2{E}_{n_1+\cdots +n_s-l,q^{-1}}^{\left(\alpha \right)}+{\left[2\right]}_q\right).\end{array}$$

(21)

From the binomial theorem and the definition of q-Bernstein polynomials, we note that

$$\begin{array}{c}{\int }_{{\mathbb{Z}}_p}\underset{s-\mathsf{\text{times}}}{\underbrace{B_{k,n_1}^{\left(\alpha \right)}\left(x,q\right)\cdots B_{k,n_s}^{\left(\alpha \right)}\left(x,q\right)}}d{\mu }_{-q}\left(x\right)\\ =\left(\begin{array}{c}\hfill n_1\hfill \\ \hfill k\hfill \end{array}\right)\cdots \left(\begin{array}{c}\hfill n_s\hfill \\ \hfill k\hfill \end{array}\right)\sum _{l=0}^{n_1+\cdots +n_s-sk}{\left(-1\right)}^l\left(\begin{array}{c}\hfill n_1+\cdots +n_s-sk\hfill \\ \hfill l\hfill \end{array}\right){\int }_{{\mathbb{Z}}_p}{\left[x\right]}_{q^{\alpha }}^{sk+l}d{\mu }_{-q}\left(x\right)\\ =\left(\begin{array}{c}\hfill n_1\hfill \\ \hfill k\hfill \end{array}\right)\cdots \left(\begin{array}{c}\hfill n_s\hfill \\ \hfill k\hfill \end{array}\right)\sum _{l=0}^{n_1+\cdots +n_s-sk}{\left(-1\right)}^l\left(\begin{array}{c}\hfill n_1+\cdots +n_s-sk\hfill \\ \hfill l\hfill \end{array}\right)E_{sk+l,q}^{\left(\alpha \right)}.\end{array}$$

(22)

Therefore, by (21) and (22), we obtain the following theorem.

Theorem 9. Let s ∈ ℕ with s ≥ 2. For n₁, n₂, ..., n _s , k ∈ ℤ₊ with n₁ + ⋯ + n _s > sk, we have

$$\begin{array}{c}\sum _{l=0}^{n_1+\cdots +n_s-sk}{\left(-1\right)}^l\left(\begin{array}{c}\hfill n_1+\cdots +n_s-sk\hfill \\ \hfill l\hfill \end{array}\right)E_{sk+l,q}^{\left(\alpha \right)}\\ =\left\{\begin{array}{cc}\hfill q^2{E}_{n_1+\cdots +n_s,q^{-1}}^{\left(\alpha \right)}+{\left[2\right]}_q,\hfill & \hfill \mathsf{\text{if}}k=0,\hfill \\ \hfill q^2{\sum }_{l=0}^{sk}\left(\begin{array}{c}\hfill sk\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{l+sk}{E}_{n_1+\cdots +n_s-l,q^{-1}}^{\left(\alpha \right)},\hfill & \hfill \mathsf{\text{if }}k>0.\hfill \end{array}\right.\end{array}$$