# 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 p , ${ℚ}_{p}$, and 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 ${ℚ}_{p}$, respectively. Let be the set of natural numbers and + = {0}. Let ν p be the normalized exponential valuation of 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 , or a p-adic number q p . If q , then one normally assumes |q| < 1, and if q 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 )

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

$\begin{array}{lll}\hfill {B}_{n,q}^{\left(\alpha \right)}\left(f|x\right)& =\sum _{k=0}^{n}\left(\begin{array}{c}\hfill n\hfill \\ \hfill k\hfill \end{array}\right)f\left(\frac{k}{n}\right){\left[x\right]}_{{q}^{\alpha }}^{k}{\left[1-x\right]}_{{q}^{-\alpha }}^{n-k}\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ =\sum _{k=0}^{n}f\left(\frac{k}{n}\right){B}_{k,n}^{\left(\alpha \right)}\left(x,q\right),.\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}$
(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( p ) be the space of continuous functions on p . For f C( p ), the fermionic q-integral on p is defined by

${I}_{q}\left(f\right)={\int }_{{ℤ}_{p}}f\left(x\right)d{\mu }_{-q}\left(x\right)=\underset{N\to \infty }{lim}\frac{1+q}{1+{q}^{{p}^{N}}}\sum _{x=0}^{{p}^{N}-1}f\left(x\right)\left(-q{\right)}^{x},$
(2)

see .

For n , by (2), we get

${q}^{n}{\int }_{{ℤ}_{p}}f\left(x+n\right)d{\mu }_{-q}\left(x\right)=\left(-{1\right)}^{n}{\int }_{{ℤ}_{p}}f\left(x\right)d{\mu }_{-q}\left(x\right)+{\left[2\right]}_{q}\sum _{l=0}^{n-1}{\left(-1\right)}^{n-1-l}{q}^{l}f\left(l\right),$
(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:

${\int }_{{ℤ}_{p}}{\left[x+y\right]}_{{q}^{\alpha }}^{n}d{\mu }_{-q}\left(y\right)={E}_{n,q}^{\left(\alpha \right)}\left(x\right),\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{for}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}n\in {ℤ}_{+}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{and}}\phantom{\rule{2.77695pt}{0ex}}\alpha \in ℤ,$
(4)

see .

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 ).

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 .

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 .

That is, (6) can be written as

${E}_{n,q}^{\left(\alpha \right)}\left(x\right)=\left({q}^{\alpha x}{E}_{q}^{\left(\alpha \right)}+{\left[x\right]}_{{q}^{\alpha }}{\right)}^{n},n\in {ℤ}_{+}.$
(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 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 and q with |q| < 1.

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

${F}_{q}\left(t,x\right)=\sum _{n=0}^{\infty }\phantom{\rule{2.77695pt}{0ex}}{E}_{n,q}^{\left(\alpha \right)}\left(x\right)\frac{{t}^{n}}{n!}.$
(8)

By (5) and (8), we get

$\begin{array}{lll}\hfill {F}_{q}\left(t,x\right)& =\sum _{n=0}^{\infty }\left(\frac{{\left[2\right]}_{q}}{{\left(1-{q}^{\alpha }\right)}^{n}}\sum _{l=0}^{n}\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{l}\frac{{q}^{\alpha lx}}{1+{q}^{\alpha l+1}}\right)\frac{{t}^{n}}{n!}\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ ={\left[2\right]}_{q}\sum _{m=0}^{\infty }{\left(-1\right)}^{m}{q}^{m}{e}^{{\left[x+m\right]}_{q}\alpha \phantom{\rule{2.77695pt}{0ex}}t}.\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}$
(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 +, we have

${E}_{0,q}^{\left(\alpha \right)}=1,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{and}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}q{E}_{n,q}^{\left(\alpha \right)}\left(1\right)+{E}_{n,q}^{\left(\alpha \right)}=0\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{if}}\phantom{\rule{2.77695pt}{0ex}}n>0.$

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

Corollary 2. For n +, we have

${E}_{0,q}^{\left(\alpha \right)}=1,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{and}}\phantom{\rule{2.77695pt}{0ex}}q{\left({q}^{\alpha }{E}_{q}^{\left(\alpha \right)}+1\right)}^{n}+{E}_{n,q}^{\left(\alpha \right)}=0\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{if}}\phantom{\rule{2.77695pt}{0ex}}n>0,$

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 +. Then we have

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

By Corollary 2, we get

$\begin{array}{lll}\hfill {q}^{2}{E}_{n,q}^{\left(\alpha \right)}\left(2\right)-{q}^{2}-q& ={q}^{2}\sum _{l=0}^{n}\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right){q}^{\alpha l}{\left({q}^{\alpha }{E}_{q}^{\left(\alpha \right)}+1\right)}^{l}-{q}^{2}-q\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ =-q\sum _{l=1}^{n}\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right){q}^{\alpha l}{E}_{l,q}^{\left(\alpha \right)}-q\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ =-q\sum _{l=0}^{n}\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right){q}^{\alpha l}{E}_{l,q}^{\left(\alpha \right)}\phantom{\rule{2em}{0ex}}& \hfill \text{(3)}\\ =-q{E}_{n,q}^{\left(\alpha \right)}\left(1\right)={E}_{n,q}^{\left(\alpha \right)}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{if}}\phantom{\rule{2.77695pt}{0ex}}n>0.\phantom{\rule{2em}{0ex}}& \hfill \text{(4)}\\ \hfill \text{(5)}\end{array}$
(12)

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

Theorem 4. For 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 p and q p with |1 - q| p < 1.

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

$\begin{array}{c}q{\int }_{{ℤ}_{p}}{\left[1-x\right]}_{{q}^{-\alpha }}^{n}d{\mu }_{-q}\left(x\right)=\left(-{1\right)}^{n}{q}^{\alpha n+1}{\int }_{{ℤ}_{p}}{\left[x-1\right]}_{{q}^{\alpha }}^{n}d{\mu }_{-q}\left(x\right)\\ =q\sum _{l=0}^{n}\left(\begin{array}{c}n\\ l\end{array}\right){\left(-1\right)}^{l}{\int }_{{ℤ}_{p}}{\left[x\right]}_{{q}^{\alpha }}^{l}d{\mu }_{-q}\left(x\right).\end{array}$
(13)

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

Theorem 5. For n +, we get

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

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

Corollary 6. For n , we have

$\begin{array}{lll}\hfill {\int }_{{ℤ}_{p}}{\left[1-x\right]}_{{q}^{-\alpha }}^{n}d{\mu }_{-q}\left(x\right)& ={E}_{n,{q}^{-1}}^{\left(\alpha \right)}\left(2\right)\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ ={q}^{2}{E}_{n,{q}^{-1}}^{\left(\alpha \right)}+{\left[2\right]}_{q}.\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}$

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

${B}_{k,n}^{\left(\alpha \right)}\left(x,q\right)=\left(\begin{array}{c}\hfill n\hfill \\ \hfill k\hfill \end{array}\right){\left[x\right]}_{{q}^{\alpha }}^{k}{\left[1-x\right]}_{{q}^{-\alpha }}^{n-k},\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{where}}\phantom{\rule{2.77695pt}{0ex}}n,k\in {ℤ}_{+},$
(14)

see .

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 

By (15), we get

$\begin{array}{lll}\hfill {\int }_{{ℤ}_{p}}{B}_{k,n}^{\left(\alpha \right)}\left(x,q\right)d{\mu }_{-q}\left(x\right)& ={\int }_{{ℤ}_{p}}{B}_{n-k,n}^{\left(\alpha \right)}\left(1-x,{q}^{-1}\right)d{\mu }_{-q}\left(x\right)\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ =\left(\begin{array}{c}\hfill n\hfill \\ \hfill k\hfill \end{array}\right)\sum _{l=0}^{k}\left(\begin{array}{c}\hfill k\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{k+l}{\int }_{{ℤ}_{p}}{\left[1-x\right]}_{{q}^{-\alpha }}^{n-l}d{\mu }_{-q}\left(x\right).\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}$
(16)

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

(17)

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

$\begin{array}{lll}\hfill {\int }_{{ℤ}_{p}}{B}_{k,n}^{\left(\alpha \right)}\left(x,q\right)d{\mu }_{-q}\left(x\right)& =\left(\begin{array}{c}\hfill n\hfill \\ \hfill k\hfill \end{array}\right){\int }_{{ℤ}_{p}}{\left[x\right]}_{{q}^{\alpha }}^{k}{\left[1-x\right]}_{{q}^{-\alpha }}^{n-k}d{\mu }_{-q}\left(x\right)\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ =\left(\begin{array}{c}\hfill n\hfill \\ \hfill k\hfill \end{array}\right)\sum _{l=0}^{n-k}\left(\begin{array}{c}\hfill n-k\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{l}{\int }_{{ℤ}_{p}}{\left[x\right]}_{{q}^{\alpha }}^{k+l}d{\mu }_{-q}\left(x\right)\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ =\left(\begin{array}{c}\hfill n\hfill \\ \hfill k\hfill \end{array}\right)\sum _{l=0}^{n-k}\left(\begin{array}{c}\hfill n-k\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{l}{E}_{l+k,q}^{\left(\alpha \right)}.\phantom{\rule{2em}{0ex}}& \hfill \text{(3)}\\ \hfill \text{(4)}\end{array}$
(18)

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

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

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

$\begin{array}{c}{\int }_{{ℤ}_{p}}{B}_{k,{n}_{1}}^{\left(\alpha \right)}\left(x,q\right){B}_{k,{n}_{2}}^{\left(\alpha \right)}\left(x,q\right)d{\mu }_{-q}\left(x\right)\\ =\left(\begin{array}{c}\hfill {n}_{1}\hfill \\ \hfill k\hfill \end{array}\right)\left(\begin{array}{c}\hfill {n}_{2}\hfill \\ \hfill k\hfill \end{array}\right)\sum _{l=0}^{2k}\left(\begin{array}{c}\hfill 2k\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{l+2k}{\int }_{{ℤ}_{p}}{\left[1-x\right]}_{{q}^{-\alpha }}^{{n}_{1}+{n}_{2}-l}d{\mu }_{-q}\left(x\right)\\ =\left(\begin{array}{c}\hfill {n}_{1}\hfill \\ \hfill k\hfill \end{array}\right)\left(\begin{array}{c}\hfill {n}_{2}\hfill \\ \hfill k\hfill \end{array}\right)\sum _{l=0}^{2k}\left(\begin{array}{c}\hfill 2k\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{l+2k}\left({q}^{2}{E}_{{n}_{1}+{n}_{2}-l,{q}^{-1}}^{\left(\alpha \right)}+{\left[2\right]}_{q}\right).\end{array}$
(19)

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

$\begin{array}{c}{\int }_{{ℤ}_{p}}{B}_{k,{n}_{1}}^{\left(\alpha \right)}\left(x,q\right){B}_{k,{n}_{2}}^{\left(\alpha \right)}\left(x,q\right)d{\mu }_{-q}\left(x\right)\\ =\left(\begin{array}{c}\hfill {n}_{1}\hfill \\ \hfill k\hfill \end{array}\right)\left(\begin{array}{c}\hfill {n}_{2}\hfill \\ \hfill k\hfill \end{array}\right)\sum _{l=0}^{{n}_{1}+{n}_{2}-2k}{\left(-1\right)}^{l}\left(\begin{array}{c}\hfill {n}_{1}+{n}_{2}-2k\hfill \\ \hfill l\hfill \end{array}\right){\int }_{{ℤ}_{p}}{\left[x\right]}_{{q}^{\alpha }}^{2k+l}d{\mu }_{-q}\left(x\right)\\ =\left(\begin{array}{c}\hfill {n}_{1}\hfill \\ \hfill k\hfill \end{array}\right)\left(\begin{array}{c}\hfill {n}_{2}\hfill \\ \hfill k\hfill \end{array}\right)\sum _{l=0}^{{n}_{1}+{n}_{2}-2k}{\left(-1\right)}^{l}\left(\begin{array}{c}\hfill {n}_{1}+{n}_{2}-2k\hfill \\ \hfill l\hfill \end{array}\right){E}_{2k+l,q}^{\left(\alpha \right)}.\end{array}$
(20)

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

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

Let s with s ≥ 2. For n1, n2, ..., n s , k + with n1 + + n s > sk, we have

$\begin{array}{c}{\int }_{{ℤ}_{p}}\underset{s-times}{\underset{⏟}{{B}_{k,{n}_{1}}^{\left(\alpha \right)}\left(x,q\right)\cdots {B}_{k,{n}_{s}}^{\left(\alpha \right)}\left(x,q\right)}}d{\mu }_{-q}\left(x\right)\\ =\left(\begin{array}{c}\hfill {n}_{1}\hfill \\ \hfill k\hfill \end{array}\right)\cdots \left(\begin{array}{c}\hfill {n}_{s}\hfill \\ \hfill k\hfill \end{array}\right){\int }_{{ℤ}_{p}}{\left[x\right]}_{{q}^{\alpha }}^{sk}{\left[1-x\right]}_{{q}^{-\alpha }}^{{n}_{1}+\cdots +{n}_{s}-sk}d{\mu }_{-q}\left(x\right)\\ =\left(\begin{array}{c}\hfill {n}_{1}\hfill \\ \hfill k\hfill \end{array}\right)\cdots \left(\begin{array}{c}\hfill {n}_{s}\hfill \\ \hfill k\hfill \end{array}\right)\sum _{l=0}^{sk}\left(\begin{array}{c}\hfill sk\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{l+sk}{\int }_{{ℤ}_{p}}{\left[1-x\right]}_{{q}^{-\alpha }}^{{n}_{1}+\cdots +{n}_{s}-l}d{\mu }_{-q}\left(x\right)\\ =\left(\begin{array}{c}\hfill {n}_{1}\hfill \\ \hfill k\hfill \end{array}\right)\cdots \left(\begin{array}{c}\hfill {n}_{s}\hfill \\ \hfill k\hfill \end{array}\right)\sum _{l=0}^{sk}\left(\begin{array}{c}\hfill sk\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{l+sk}\left({q}^{2}{E}_{{n}_{1}+\cdots +{n}_{s}-l,{q}^{-1}}^{\left(\alpha \right)}+{\left[2\right]}_{q}\right).\end{array}$
(21)

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

$\begin{array}{c}{\int }_{{ℤ}_{p}}\underset{s-\mathsf{\text{times}}}{\underset{⏟}{{B}_{k,{n}_{1}}^{\left(\alpha \right)}\left(x,q\right)\cdots {B}_{k,{n}_{s}}^{\left(\alpha \right)}\left(x,q\right)}}d{\mu }_{-q}\left(x\right)\\ =\left(\begin{array}{c}\hfill {n}_{1}\hfill \\ \hfill k\hfill \end{array}\right)\cdots \left(\begin{array}{c}\hfill {n}_{s}\hfill \\ \hfill k\hfill \end{array}\right)\sum _{l=0}^{{n}_{1}+\cdots +{n}_{s}-sk}{\left(-1\right)}^{l}\left(\begin{array}{c}\hfill {n}_{1}+\cdots +{n}_{s}-sk\hfill \\ \hfill l\hfill \end{array}\right){\int }_{{ℤ}_{p}}{\left[x\right]}_{{q}^{\alpha }}^{sk+l}d{\mu }_{-q}\left(x\right)\\ =\left(\begin{array}{c}\hfill {n}_{1}\hfill \\ \hfill k\hfill \end{array}\right)\cdots \left(\begin{array}{c}\hfill {n}_{s}\hfill \\ \hfill k\hfill \end{array}\right)\sum _{l=0}^{{n}_{1}+\cdots +{n}_{s}-sk}{\left(-1\right)}^{l}\left(\begin{array}{c}\hfill {n}_{1}+\cdots +{n}_{s}-sk\hfill \\ \hfill l\hfill \end{array}\right){E}_{sk+l,q}^{\left(\alpha \right)}.\end{array}$
(22)

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

Theorem 9. Let s with s ≥ 2. For n1, n2, ..., n s , k + with n1 + + n s > sk, we have

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## 6. Acknowledgements

The authors would like to express their sincere gratitude to referee for his/her valuable comments.

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Correspondence to Taekyun Kim.

### 4. Competing interests

The authors declare that they have no competing interests. Acknowledgment The authors would like to express their sincere gratitude to referee for his/her valuable comments.

### 5. Authors' contributions

All authors contributed equally to the manuscript and read and approved the finial manuscript.

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Kim, T., Kim, YH. & Ryoo, C.S. Some identities on the weighted q-Euler numbers and q-Bernstein polynomials. J Inequal Appl 2011, 64 (2011). https://doi.org/10.1186/1029-242X-2011-64

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• DOI: https://doi.org/10.1186/1029-242X-2011-64

### Keywords

• Euler numbers and polynomials
• q-Euler numbers and polynomials
• weighted
• q-Euler numbers and polynomials
• Bernstein polynomials
• q-Bernstein polynomials 