# The twisted (h, q)-Genocchi numbers and polynomials with weight α and q-bernstein polynomials with weight α

## Abstract

In this article, we give some identities on the twisted (h, q)-Genocchi numbers and polynomials and q-Bernstein polynomials with weighted α.

## 1 Introduction

Let p be a fixed odd prime number. The symbol, ${ℤ}_{p},\phantom{\rule{0.3em}{0ex}}{ℚ}_{p}$, and ${ℂ}_{p}$ denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of algebraic closure of ${ℚ}_{p}$, respectively. Let be the set of natural numbers and ${ℤ}_{+}=ℕ\cup \left\{0\right\}$. As well known definition, the p-adic absolute value is given by |x| p = p-r , where $x={p}^{r}\frac{t}{s}$ with (t, p) = (s, p) = (t, s) = 1. When one talks of q-extension, q is variously considered as an indeterminate, a complex number $q\in ℂ$, or a p-adic number $q\in {ℂ}_{p}$. In this article, we assume that $q\in {ℂ}_{p}$ with |1 - q| p < 1.

For $f\in UD\left({ℤ}_{p}\right)=\left\{f/f:{ℤ}_{p}\to {ℂ}_{p}$ is uniformly differentiable function}, Kim defined the fermionic p-adic q-integral on ${ℤ}_{p}$ as follows:

${I}_{-q}\left(f\right)=\underset{{ℤ}_{p}}{\int }f\left(x\right)d{\mu }_{-q}\left(x\right)=\underset{N\to \infty }{\text{lim}}\frac{1}{{\left[{p}^{N}\right]}_{-q}}\sum _{x=0}^{{p}^{N}-1}f\left(x\right){\left(-q\right)}^{x}.$
(1.1)

For $n\in ℕ$, let f n (x) = f(x + n) be translation. As well known equation, by (1.1), we have

${q}^{n}{I}_{-q}\left({f}_{n}\right)={\left(-1\right)}^{n}{I}_{-q}\left(f\right)+{\left[2\right]}_{q}\sum _{l-0}^{n-1}{\left(-1\right)}^{n-1-l}{q}^{l}f\left(l\right),.$
(1.2)

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

limq→1|x| q = x for any x with |x| p 1 in the present p-adic case. To investigate relation of the twisted (h, q)-Genocchi numbers and polynomials with weight α and the q-Bernstein polynomials with weight α, we will use useful property for ${\left[x\right]}_{{q}^{\alpha }}$ as following;

$\begin{array}{c}{\left[x\right]}_{{q}^{\alpha }}=1-{\left[1-x\right]}_{{q}^{-\alpha }}\\ {\left[1-x\right]}_{{q}^{-\alpha }}=1-{\left[x\right]}_{{q}^{\alpha }}\end{array}$
(1.3)

The twisted (h, q)-Genocchi numbers and polynomials with weight α are defined by the generating function, respectively:

${G}_{n,q,w}^{\left(h,\alpha \right)}=n\underset{{ℤ}_{p}}{\int }{q}^{x\left(h-1\right)}{\varphi }_{w}\left(x\right){\left[x\right]}_{{q}^{\alpha }}^{n-1}d{\mu }_{-q}\left(x\right).$
(1.4)
${G}_{n,q,w}^{\left(h,\alpha \right)}\left(x\right)=n\underset{{ℤ}_{p}}{\int }{q}^{y\left(h-1\right)}{\varphi }_{w}\left(y\right){\left[y+x\right]}_{{q}^{\alpha }}^{n-1}d{\mu }_{-q}\left(y\right).$
(1.5)

In the special case, $x=0,\phantom{\rule{0.3em}{0ex}}{G}_{n,q,w}^{\left(h,\alpha \right)}\left(0\right)={G}_{n,q,w}^{\left(h,\alpha \right)}$ are called the n th twisted (h, q)-Genocchi numbers with weight α (see ).

Let ${C}_{{p}^{n}}=\left\{w|{w}^{{p}^{n}}=1\right\}$ be the cyclic group of order pn and let

${T}_{p}=\underset{n\to \infty }{\text{lim}}{C}_{{p}^{n}}={\cup }_{n\ge 1}{C}_{{p}^{n}}$

see .

Kim defined the q-Bernstein polynomials with weight α of degree n as follows:

(1.6)

cf .

In this article, we investigate some properties for the twisted (h, q)-Genocchi numbers and polynomials with weight α. By using these properties, we give some interesting identities on the twisted (h, q)-Genocchi polynomials with weight α and q-Bernstein polynomials with weight α.

## 2 Twisted (h, q)-genocchi numbrs and polynomials with weight α and q-bernstein polynomials with weight α

From (1.2), we can get the following form for the twisted (h, q)-Genocchi numbers with weight α:

${G}_{0,q,w}^{\left(h,\alpha \right)}=0,\phantom{\rule{0.3em}{0ex}}\text{and}\phantom{\rule{0.3em}{0ex}}{q}^{h}w{G}_{n,q,w}^{\left(h,\alpha \right)}\left(1\right)+{G}_{n,q,w}^{\left(h,\alpha \right)}=\left\{\begin{array}{cc}\hfill {\left[2\right]}_{q,}\hfill & \hfill \text{if}\phantom{\rule{0.3em}{0ex}}n=1,\hfill \\ \hfill 0,\hfill & \hfill \text{if}\phantom{\rule{0.3em}{0ex}}n>1,\hfill \end{array}\right\$
(2.1)
${G}_{0,q,w}^{\left(h,\alpha \right)}=0,\phantom{\rule{0.3em}{0ex}}\text{and}\phantom{\rule{0.3em}{0ex}}{q}^{h}w{\left(1+{q}^{\alpha }{G}_{q,w}^{\left(h,\alpha \right)}\right)}^{n}+{q}^{\alpha }{G}_{n,q,w}^{\left(h,\alpha \right)}=\left\{\begin{array}{cc}\hfill {q}^{\alpha }{\left[2\right]}_{q,}\hfill & \hfill \text{if}\phantom{\rule{0.3em}{0ex}}n=1,\hfill \\ \hfill 0,\hfill & \hfill \text{if}\phantom{\rule{0.3em}{0ex}}n>1,\hfill \end{array}\right\$
(2.2)
${q}^{\alpha x}{G}_{n+1,q,w}^{\left(h,\alpha \right)}\left(x\right)={\left({\left[x\right]}_{q}^{\alpha }+{q}^{\alpha x}{G}_{q,w}^{\left(h,\alpha \right)}\right)}^{n+1}$
(2.3)

with usual convention about replacing ${\left({G}_{q,w}^{\left(h,\alpha \right)}\right)}^{n}$ by ${G}_{n,q,w}^{\left(h,\alpha \right)}$.

By (1.4), we can obtain

${G}_{n,q,w}^{\left(h,\alpha \right)}\left(x\right)=n{\left[2\right]}_{q}{\left(\frac{1}{1-{q}^{\alpha }}\right)}^{n-1}\sum _{l=0}^{n-1}\left(\begin{array}{c}\hfill n-1\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{l}\frac{1}{1+w{q}^{\alpha l+h}}$
(2.4)

By (2.4), we can get

$\begin{array}{ll}\hfill {G}_{n,{q}^{-1},{w}^{-1}}^{\left(h,\alpha \right)}\left(1-x\right)& =n{\left[2\right]}_{{q}^{-1}}{\left(\frac{1}{1-{q}^{-\alpha }}\right)}^{n-1}\sum _{l=0}^{n-1}\left(\begin{array}{c}\hfill n-1\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{l}{\left({q}^{-1}\right)}^{\alpha l\left(1-x\right)}\frac{1}{1+{w}^{-1}{\left({q}^{-1}\right)}^{\alpha l+h}}\phantom{\rule{2em}{0ex}}\\ =n\frac{1}{q}{\left[2\right]}_{q}{\left(\frac{1}{1-{q}^{\alpha }}\right)}^{n-1}{\left(-1\right)}^{n-1}{q}^{\alpha n+\alpha }\sum _{l=0}^{n-1}\left(\begin{array}{c}\hfill n-1\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{l}{q}^{\alpha lx}\frac{w{q}^{h}}{1+w{q}^{\alpha l+h}}\phantom{\rule{2em}{0ex}}\\ =n{\left[2\right]}_{q}{\left(\frac{1}{1-{q}^{\alpha }}\right)}^{n-1}\sum _{l=0}^{n-1}\left(\begin{array}{c}\hfill n-1\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{l}{q}^{\alpha lx}\frac{1}{1+w{q}^{\alpha l+h}}\frac{1}{q}{q}^{\alpha n-\alpha }{\left(-1\right)}^{n-1}w{q}^{h}\phantom{\rule{2em}{0ex}}\\ ={\left(-1\right)}^{n-1}w{q}^{\alpha \left(n-1\right)+\left(h-1\right)}{G}_{n,q,w}^{\left(h,\alpha \right)}\left(x\right).\phantom{\rule{2em}{0ex}}\end{array}$

So, we get the following theorem.

Theorem 1. Let $n\in {ℤ}_{+}$. For w T p , we have

${G}_{n,q,w}^{\left(h,\alpha \right)}\left(x\right)={\left(-1\right)}^{n-1}{w}^{-1}{q}^{\alpha \left(1-n\right)+\left(1-h\right)}{G}_{n,{q}^{-1},{w}^{-1}}^{\left(h,\alpha \right)}\left(1-x\right).$

By (2.1), (2.2), and (2.3), we note that

$\begin{array}{ll}\hfill {G}_{n,q,w}^{\left(h,\alpha \right)}& =-w{q}^{h}{G}_{n,q,w}^{\left(h,\alpha \right)}\left(1\right)\phantom{\rule{2em}{0ex}}\\ =-w{q}^{h}\left({q}^{-\alpha }{\left(1+{g}^{\alpha }{G}_{q,w}^{\left(h,\alpha \right)}\right)}^{n}\right)\phantom{\rule{2em}{0ex}}\\ =-w{q}^{h-\alpha }\sum _{l=0}^{n}\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right){\left({q}^{\alpha }\right)}^{l}{G}_{l,q,w}^{\left(h,\alpha \right)}\phantom{\rule{2em}{0ex}}\\ =-w{q}^{h}\left(\begin{array}{c}\hfill n\hfill \\ \hfill 1\hfill \end{array}\right){G}_{1,q,w}^{\left(h,\alpha \right)}-w{q}^{h-\alpha }\sum _{l=2}^{n}\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right){q}^{\alpha l}\left(-w{q}^{h}{G}_{l,q,w}^{\left(h,\alpha \right)}\left(1\right)\right)\phantom{\rule{2em}{0ex}}\\ =-w{q}^{h}\left(\begin{array}{c}\hfill n\hfill \\ \hfill 1\hfill \end{array}\right){G}_{1,q,w}^{\left(h,\alpha \right)}-w{q}^{h-\alpha }\sum _{l=2}^{n}\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right){q}^{\alpha l}\left(-w{q}^{h}{q}^{-\alpha }{\left(1+{q}^{\alpha }{G}_{q,w}^{\left(h,\alpha \right)}\right)}^{l}\right)\phantom{\rule{2em}{0ex}}\\ =-nw{q}^{h}{G}_{1,q,w}^{\left(h,\alpha \right)}+{w}^{2}{q}^{2h-2\alpha }\sum _{l=2}^{n}\left(\begin{array}{c}\hfill n\hfill \\ \hfill l\hfill \end{array}\right){q}^{\alpha l}{\left(1+{q}^{\alpha }{G}_{q,w}^{\left(h,\alpha \right)}\right)}^{l}\phantom{\rule{2em}{0ex}}\\ =-nw{q}^{h}{G}_{1,q,w}^{\left(h,\alpha \right)}+{w}^{2}{q}^{2h-2\alpha }{\left(1+{q}^{\alpha }\left(1+{g}^{\alpha }{G}_{q,w}^{\left(h,\alpha \right)}\right)\right)}^{n}-n{w}^{2}{q}^{2h-2\alpha }{q}^{\alpha }{\left(1+{q}^{\alpha }{G}_{q,w}^{\left(h,\alpha \right)}\right)}^{1}\phantom{\rule{2em}{0ex}}\\ =-nw{q}^{h}{G}_{1,q,w}^{\left(h,\alpha \right)}+{w}^{2}{q}^{2h-2\alpha }{\left({\left[2\right]}_{{q}^{\alpha }}+{q}^{2\alpha }{G}_{q,w}^{\left(h,\alpha \right)}\right)}^{n}-n{w}^{2}{q}^{2h-\alpha }\left({q}^{\alpha }{G}_{1,q,w}^{\left(h,\alpha \right)}\right)\phantom{\rule{2em}{0ex}}\\ =-nw{q}^{h}{G}_{1,q,w}^{\left(h,\alpha \right)}+{w}^{2}{q}^{2h-2\alpha }{\left({\left[2\right]}_{{q}^{\alpha }}+{q}^{2\alpha }{G}_{q,w}^{\left(h,\alpha \right)}\right)}^{n}-n{w}^{2}{q}^{2h}{G}_{1,q,w}^{\left(h,\alpha \right)}\phantom{\rule{2em}{0ex}}\\ =-nw{q}^{h}{G}_{1,q,w}^{\left(h,\alpha \right)}+{w}^{2}{q}^{2h}{G}_{n,q,w}^{\left(h,\alpha \right)}\left(2\right)-n{w}^{2}{q}^{2h}{G}_{1,q,w}^{\left(h,\alpha \right)}\phantom{\rule{2em}{0ex}}\end{array}$
(2.5)

Therefore, by (2.5), we obtain the theorem below.

Theorem 2. For $n\in ℕ$ with n > 1, we have

${G}_{n,q,w}^{\left(h,\alpha \right)}\left(2\right)={w}^{-2}{q}^{-2h}{G}_{n,q,w}^{\left(h,\alpha \right)}+{w}^{-1}{q}^{-h}\frac{n{\left[2\right]}_{q}}{1+w{q}^{h}}+\frac{n{\left[2\right]}_{q}}{1+w{q}^{h}}.$

From Theorem 2,

$\begin{array}{ll}\hfill \frac{{G}_{n+1,q,w}^{\left(h,\alpha \right)}\left(2\right)}{n+1}& =\frac{1}{n+1}\left(\frac{\left(n+1\right){\left[2\right]}_{q}}{1+w{q}^{h}}+\frac{\left(n+1\right){w}^{-1}{q}^{-h}{\left[2\right]}_{q}}{1+w{q}^{h}}\right)+{w}^{-2}{q}^{-2h}\frac{{G}_{n+1,q,w}^{\left(h,\alpha \right)}}{n+1}\phantom{\rule{2em}{0ex}}\\ =\frac{{\left[2\right]}_{q}}{1+w{q}^{h}}+{w}^{-1}{q}^{-h}\frac{{\left[2\right]}_{q}}{1+w{q}^{h}}+{w}^{-2}{q}^{-2h}\frac{{G}_{n+1,q,w}^{\left(h,\alpha \right)}}{n+1}\phantom{\rule{2em}{0ex}}\end{array}$

Therefore, we obtain the Corollary 3 by (1.5) and Theorem 2.

Corollary 3. For $n\in ℕ$, we have

$\underset{{ℤ}_{p}}{\int }{q}^{y\left(h-1\right)}{\varphi }_{w}\left(y\right){\left[y+2\right]}_{{q}^{\alpha }}^{n}q{\mu }_{-q}\left(y\right)=\frac{{\left[2\right]}_{q}}{1+w{q}^{h}}+{w}^{-1}{q}^{-h}\frac{{\left[2\right]}_{q}}{1+w{q}^{h}}+{w}^{-2}{q}^{-2h}\frac{{G}_{n+1,q,w}^{\left(h,\alpha \right)}}{n+1}$

By Theorems 1, 2 and fermionic integral on ${ℤ}_{p}$, we note that

$\begin{array}{ll}\hfill \underset{{ℤ}_{p}}{\int }{q}^{x\left(h-1\right)}{\varphi }_{w}\left(x\right){\left[1-x\right]}_{{q}^{-\alpha }}^{n}q{\mu }_{-q}\left(x\right)& ={\left(-1\right)}^{n}{q}^{\alpha n}\underset{{ℤ}_{p}}{\int }{q}^{x\left(h-1\right)}{\varphi }_{w}\left(x\right){\left[x-1\right]}_{{q}^{\alpha }}^{n}d{\mu }_{-q}\left(x\right)\phantom{\rule{2em}{0ex}}\\ ={\left(-1\right)}^{n}{q}^{\alpha n}\frac{{G}_{n+1,q,w}^{\left(h,\alpha \right)}\left(-1\right)}{n+1}\phantom{\rule{2em}{0ex}}\\ ={w}^{-1}{q}^{1-h}\frac{{G}_{n+1,{q}^{-1},{w}^{-1}}^{\left(h,\alpha \right)}\left(2\right)}{n+1}\phantom{\rule{2em}{0ex}}\\ ={w}^{-1}{q}^{1-h}\left(\frac{{\left[2\right]}_{{q}^{-1}}}{1+{q}^{-h}{w}^{-1}}+w{q}^{h}\frac{{\left[2\right]}_{{q}^{-1}}}{1+{q}^{-1}{w}^{-1}}+{w}^{2}{q}^{2h}\frac{{G}_{n+1,{q}^{-1},{w}^{-1}}^{\left(h,\alpha \right)}}{n+1}\right)\phantom{\rule{2em}{0ex}}\\ =\frac{{\left[2\right]}_{q}}{1+w{q}^{h}}+w{q}^{h}\frac{{\left[2\right]}_{q}}{1+w{q}^{h}}+w{q}^{h+1}\frac{{G}_{n+1,{q}^{-1},{w}^{-1}}^{\left(h,\alpha \right)}}{n+1}\phantom{\rule{2em}{0ex}}\\ ={\left[2\right]}_{q}+w{q}^{h+1}\frac{{G}_{n+1,{q}^{-1},{w}^{-1}}^{\left(h,\alpha \right)}}{n+1}.\phantom{\rule{2em}{0ex}}\end{array}$
(2.6)

Hence, we get the following theorem.

Theorem 4. for $n\in ℕ$ with n > 1, we have

$\underset{{ℤ}_{p}}{\int }{q}^{x\left(h-1\right)}{\varphi }_{w}\left(x\right){\left[1-x\right]}_{{q}^{-\alpha }}^{n}d{\mu }_{-q}\left(x\right)={\left[2\right]}_{q}+w{q}^{h+1}\frac{{G}_{n+1,{q}^{-1},{w}^{-1}}^{\left(h,\alpha \right)}}{n+1}.$
(2.7)

Corollary 5.

From (1.3) and Theorem 4, we take the fermionic p-adic invariant integral on ${ℤ}_{p}$ for q-Bernstein polynomials as follows:

$\begin{array}{ll}\hfill \underset{{ℤ}_{p}}{\int }{q}^{x\left(h-1\right)}{\varphi }_{w}\left(x\right){B}_{k,n}\left(x,q\right)d{\mu }_{-q}\left(x\right)& =\underset{{ℤ}_{p}}{\int }{q}^{x\left(h-1\right)}{\varphi }_{w}\left(x\right)\left(\begin{array}{c}\hfill n\hfill \\ \hfill k\hfill \end{array}\right){\left[x\right]}_{{q}^{\alpha }}^{k}{\left[1-x\right]}_{q-\alpha }^{n-k}d{\mu }_{-q}\left(x\right)\phantom{\rule{2em}{0ex}}\\ =\left(\begin{array}{c}\hfill n\hfill \\ \hfill k\hfill \end{array}\right)\underset{{ℤ}_{p}}{\int }{q}^{x\left(h-1\right)}{\varphi }_{w}\left(x\right){\left[x\right]}_{{q}^{\alpha }}^{k}{\left(1-{\left[x\right]}_{{q}^{\alpha }}\right)}^{n-k}d{\mu }_{-q}\left(x\right)\phantom{\rule{2em}{0ex}}\\ =\left(\begin{array}{c}\hfill n\hfill \\ \hfill k\hfill \end{array}\right)\underset{{ℤ}_{p}}{\int }{q}^{x\left(h-1\right)}{\varphi }_{w}\left(x\right){\left[x\right]}_{{q}^{\alpha }}^{k}{\left(1-{\left[x\right]}_{{q}^{\alpha }}\right)}^{n-k}d{\mu }_{-q}\left(x\right)\phantom{\rule{2em}{0ex}}\\ =\left(\begin{array}{c}\hfill n\hfill \\ \hfill k\hfill \end{array}\right)\sum _{l=0}^{n-k}\left(\begin{array}{c}\hfill n\hfill \\ \hfill n-k\hfill \end{array}\right){\left(-1\right)}^{l}\underset{{ℤ}_{p}}{\int }{q}^{x\left(h-1\right)}{\varphi }_{w}\left(x\right){\left[x\right]}_{{q}^{\alpha }}^{k+l}d{\mu }_{-q}\left(x\right)\phantom{\rule{2em}{0ex}}\\ =\left(\begin{array}{c}\hfill n\hfill \\ \hfill k\hfill \end{array}\right)\sum _{l=0}^{n-k}\left(\begin{array}{c}\hfill n-k\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{l}\frac{{G}_{k+l+1,q,w}^{\left(h,\alpha \right)}}{k+l+1}\phantom{\rule{2em}{0ex}}\end{array}$
(2.8)

And we get the following formula;

$\begin{array}{c}\underset{{ℤ}_{p}}{\int }{q}^{x\left(h-1\right)}{\varphi }_{w}\left(x\right){B}_{k,n}\left(x,q\right)d{\mu }_{-q}\left(x\right)\\ =\underset{{ℤ}_{p}}{\int }{q}^{x\left(h-1\right)}{\varphi }_{w}\left(x\right)\left(\begin{array}{c}n\\ k\end{array}\right){\left[x\right]}_{{q}^{\alpha }}^{n-k}{\left[1-x\right]}_{{q}^{-\alpha }}^{k}d{\mu }_{-q}\left(x\right)\\ =\underset{{ℤ}_{p}}{\int }{q}^{x\left(h-1\right)}{\varphi }_{w}\left(x\right)\left(\begin{array}{c}n\\ k\end{array}\right){\left[1-x\right]}_{{q}^{-\alpha }}^{k}{\left(1-{\left[1-x\right]}_{{q}^{-\alpha }}\right)}^{n-k}d{\mu }_{-q}\left(x\right)\\ =\left(\begin{array}{c}n\\ k\end{array}\right)\sum _{l=0}^{n-k}\left(\begin{array}{c}\hfill n-k\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{n-k-l}{\left[1-x\right]}_{{q}^{-\alpha }}^{n-k-l}\underset{{ℤ}_{p}}{\int }{q}^{x\left(h-1\right)}{\varphi }_{w}\left(x\right){\left[1-x\right]}_{{q}^{-\alpha }}^{k}d{\mu }_{-q}\left(x\right)\\ =\left(\begin{array}{c}n\\ k\end{array}\right)\sum _{l=0}^{n-k}\left(\begin{array}{c}\hfill n-k\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{n-k-l}\underset{{ℤ}_{p}}{\int }{q}^{x\left(h-1\right)}{\varphi }_{w}\left(x\right){\left[1-x\right]}_{{q}^{-\alpha }}^{n-l}d{\mu }_{-q}\left(x\right)\\ =\left(\begin{array}{c}n\\ k\end{array}\right)\sum _{l=0}^{n-k}\left(\begin{array}{c}\hfill n-k\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{n-k-l}\left({\left[2\right]}_{q}+w{q}^{1+h}\frac{{G}_{n-l+1,{q}^{-1},{w}^{-1}}^{\left(h,\alpha \right)}}{n-l+1}\right)\end{array}$
(2.9)

Hence, we can get the following theorem by (2.8) and (2.9).

Theorem 5. for $n\in ℕ$ with n > 1, we have

$\begin{array}{c}\sum _{l=0}^{n-k}\left(\begin{array}{c}\hfill n-k\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{l}\frac{{G}_{k+l+1,q,w}^{\left(h,\alpha \right)}}{k+l+1}\\ =\sum _{l=0}^{n-k}\left(\begin{array}{c}\hfill n-k\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{n-k-l}\left(\left[2\right]+w{q}^{1+h}\frac{{G}_{n-l+1,{q}^{-1},{w}^{-1}}^{\left(h,\alpha \right)}}{n-l+1}\right)\\ =\sum _{l=0}^{n-k}\left(\begin{array}{c}\hfill n-k\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{n-k-l}\left({w}^{-1}{q}^{1-h}\frac{{G}_{n-l+1,{q}^{-1},{w}^{-1}}^{\left(h,\alpha \right)}\left(2\right)}{n-l+1}\right)\end{array}$
(2.10)

Also, we can see that

$\begin{array}{l}\underset{{ℤ}_{p}}{\int }{q}^{x\left(h-1\right)}\varphi w\left(x\right){B}_{k,n}\left(x,q\right)d{\mu }_{q}\left(x\right)\phantom{\rule{2em}{0ex}}\\ =\left(\begin{array}{c}n\\ k\end{array}\right)\sum _{l=0}^{n-k}\left(\begin{array}{c}\hfill n-k\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{l}\frac{{G}_{k+l+1,q,w}^{\left(h,\alpha \right)}}{k+l+1}\phantom{\rule{2em}{0ex}}\\ =\left(\begin{array}{c}n\\ k\end{array}\right)\underset{l=0}{\int }{q}^{x\left(h-1\right)}{\varphi }_{w}\left(x\right){\left[1-x\right]}_{{q}^{-\alpha }}^{n-k}{\left[x\right]}_{{q}^{\alpha }}^{k}d{\mu }_{-q}\left(x\right)\phantom{\rule{2em}{0ex}}\\ =\left(\begin{array}{c}n\\ k\end{array}\right)\underset{l=0}{\int }{q}^{x\left(h-1\right)}{\varphi }_{w}\left(x\right){\left[1-x\right]}_{{q}^{-\alpha }}^{n-k}\left(1-\left[1-x\right]{q}^{-\alpha }\right)d{\mu }_{-q}\left(x\right)\phantom{\rule{2em}{0ex}}\\ =\left(\begin{array}{c}n\\ k\end{array}\right)\sum _{l=0}^{k}\left(\begin{array}{c}k\\ l\end{array}\right){\left(-1\right)}^{k-1}\underset{{ℤ}_{p}}{\int }{q}^{x\left(h-1\right)}{\varphi }_{w}\left(x\right){\left[1-x\right]}_{{q}^{-\alpha }}^{n-k}d{\mu }_{-q}\left(x\right)\phantom{\rule{2em}{0ex}}\\ =\left(\begin{array}{c}n\\ k\end{array}\right)\sum _{l=0}^{k}\left(\begin{array}{c}k\\ l\end{array}\right){\left(-1\right)}^{k-1}\left({\left[2\right]}_{q}+w{q}^{1+h}\frac{{G}_{n-l+1,{q}^{-1},{w}^{-1}}^{\left(h,\alpha \right)}}{n-l+1}\right).\phantom{\rule{2em}{0ex}}\end{array}$
(2.11)

Therefore, we have the theorem below.

Theorem 6. For $n,k\in {ℤ}_{+}$ with n > k + 1, we have

$\begin{array}{c}\underset{{ℤ}_{p}}{\int }{q}^{x\left(h-1\right)}{\varphi }_{w}\left(x\right){B}_{k,n}\left(x,q\right)d{\mu }_{-q}\left(x\right)\\ =\left(\begin{array}{c}n\\ k\end{array}\right)\sum _{l=0}^{k}\left(\begin{array}{c}k\\ l\end{array}\right){\left(-1\right)}^{k-l}\left({\left[2\right]}_{q}+w{q}^{1+h}\frac{{G}_{n-l+1,{q}^{-1},{w}^{-1}}^{\left(h,\alpha \right)}}{n-l+1}\right).\end{array}$
(2.12)

By (2.7) and Theorem 6, we can get the theorem below.

Theorem 7. Let $n,k\in {ℤ}_{+}$ with n > k + 1. Then we have

$\begin{array}{c}\sum _{l=0}^{n-k}\left(\begin{array}{c}\hfill n-k\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{l}\frac{{G}_{k+l+1,q,w}^{\left(h,\alpha \right)}}{k+l+1}\\ =\sum _{l=0}^{k}\left(\begin{array}{c}k\\ l\end{array}\right){\left(-1\right)}^{k-1}\left({\left[2\right]}_{q}+w{q}^{1+h}\frac{{G}_{n-l+1,{q}^{-1},{w}^{-1}}^{\left(h,\alpha \right)}}{n-l+1}\right).\end{array}$

Let ${n}_{1},{n}_{2},k\in {ℤ}_{+}$ with n1 + n2> 2k + 1. Then we get

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

Therefore, we obtain the theorem below.

Theorem 8. For ${n}_{1},{n}_{2},k\in {ℤ}_{+}$, we have

$\begin{array}{c}\underset{{ℤ}_{p}}{\int }{q}^{x\left(h-1\right)}{\varphi }_{w}\left(x\right){B}_{k,{n}_{1}}^{\left(\alpha \right)}\left(x,q\right){B}_{k,{n}_{2}}^{\left(\alpha \right)}\left(x,q\right)d{\mu }_{-q}\left(x\right)\\ =\left(\begin{array}{c}\hfill {n}_{1}\hfill \\ \hfill k\hfill \end{array}\right)\left(\begin{array}{c}\hfill {n}_{2}\hfill \\ \hfill k\hfill \end{array}\right)\sum _{l=0}^{2k}\left(\begin{array}{c}\hfill 2k\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{2k-l}\left({\left[2\right]}_{q}+w{q}^{1+h}\frac{{G}_{{n}_{1}+{n}_{2}-l+1,{q}^{-1},{w}^{-1}}^{\left(h,\alpha \right)}}{{n}_{1}+{n}_{2}-l+1}\right)\\ =\left\{\begin{array}{cc}\hfill \left({\left[2\right]}_{q}+w{q}^{1+h}\frac{{G}_{{n}_{1}+{n}_{2}-l+1,{q}^{-1},{w}^{-1}}^{\left(h,\alpha \right)}}{{n}_{1}+{n}_{2}-l+1}\right),\hfill & \hfill \text{if}\phantom{\rule{2.77695pt}{0ex}}k=0,\hfill \\ \hfill w{q}^{1+h}\left(\begin{array}{c}\hfill {n}_{1}\hfill \\ \hfill k\hfill \end{array}\right)\left(\begin{array}{c}\hfill {n}_{2}\hfill \\ \hfill k\hfill \end{array}\right){\sum }_{l=0}^{2k}\left(\begin{array}{c}\hfill 2k\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{2k-l}\frac{{G}_{{n}_{1}+{n}_{2}-l+1,{q}^{-1},{w}^{-1}}^{\left(h,\alpha \right)}}{{n}_{1}+{n}_{2}-l+1},\hfill & \hfill \text{if}\phantom{\rule{2.77695pt}{0ex}}k>0,\hfill \end{array}\right\\end{array}$
(2.13)

And we can easily have that

$\begin{array}{c}\underset{{ℤ}_{p}}{\int }{q}^{x\left(h-1\right)}{\varphi }_{\omega }\left(x\right){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)\\ =\underset{{ℤ}_{p}}{\int }{q}^{x\left(h-1\right)}{\varphi }_{\omega }\left(x\right)\left(\begin{array}{c}\hfill {n}_{1}\hfill \\ \hfill k\hfill \end{array}\right){\left[x\right]}_{{q}^{\alpha }}^{k}{\left[1-x\right]}_{{q}^{-\alpha }}^{{n}_{1}-k}\left(\begin{array}{c}\hfill {n}_{2}\hfill \\ \hfill k\hfill \end{array}\right){\left[x\right]}_{{q}^{\alpha }}^{k}{\left[1-x\right]}_{{q}^{-\alpha }}^{{n}_{2}-k}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)\underset{{ℤ}_{p}}{\int }{q}^{x\left(h-1\right)}{\varphi }_{w}\left(x\right){\left[x\right]}_{{q}^{\alpha }}^{2k}{\left[1-x\right]}_{{q}^{-\alpha }}^{{n}_{1}+{n}_{2}-k}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)\underset{{ℤ}_{p}}{\int }{q}^{x\left(h-1\right)}{\varphi }_{w}\left(x\right){\left[x\right]}_{{q}^{\alpha }}^{2k}{\left(1-{\left[x\right]}_{{q}^{\alpha }}\right)}^{{n}_{1}+{n}_{2}-2k}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)\underset{{ℤ}_{p}}{\int }{q}^{x\left(h-1\right)}{\varphi }_{w}\left(x\right){\left[x\right]}_{{q}^{\alpha }}^{2k}\sum _{l=0}^{{n}_{1}+{n}_{2}-2k}\left(\begin{array}{c}{n}_{1}+{n}_{2}-2k\\ l\end{array}\right){\left(-1\right)}^{l}{\left[x\right]}_{{q}^{\alpha }}^{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}{n}_{1}+{n}_{2}-2k\\ l\end{array}\right)\underset{{ℤ}_{p}}{\int }{q}^{x\left(h-1\right)}{\varphi }_{w}\left(x\right){\left[x\right]}_{{q}^{\alpha }}^{2k+1}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}{n}_{1}+{n}_{2}-2k\\ l\end{array}\right)\frac{{G}_{2k+l+1,q,w}^{\left(h,\alpha \right)}}{2k+l+1},\mathsf{\text{where}}\phantom{\rule{2.77695pt}{0ex}}{n}_{\mathsf{\text{1}}},{n}_{2},k\in {ℤ}_{+}.\end{array}$
(2.14)

Therefore, by (2.14) and Theorem 8, we obtain the theorem below.

Theorem 9. Let ${n}_{1},{n}_{2},k\in {ℤ}_{+}$ with n1 + n2> 2k + 1. Then we have

$\begin{array}{c}\sum _{l=0}^{2k}\left(\begin{array}{c}\hfill 2k\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{2k-l}\left({\left[2\right]}_{q}+w{q}^{1+h}\frac{{G}_{{n}_{1}+{n}_{2}-l+1,{q}^{-1},{w}^{-1}}^{\left(h,\alpha \right)}}{{n}_{1}+{n}_{2}-l+1}\right)\\ =\sum _{l=0}^{{n}_{1}+{n}_{2}-2k}{\left(-1\right)}^{l}\left(\begin{array}{c}{n}_{1}+{n}_{2}-2k\\ l\end{array}\right)\frac{{G}_{2k+l+1,q,w}^{\left(h,\alpha \right)}}{2k+l+1}.\end{array}$

For ${n}_{1},{n}_{2},\dots ,{n}_{s},k\in {ℤ}_{+},{n}_{1}+{n}_{2}+\cdots +{n}_{s}>sk+1$, then by the symmetry of q-Bernstein polynomials with weight α, we see that

$\begin{array}{l}\underset{{ℤ}_{p}}{\int }{q}^{x\left(h-1\right)}{\varphi }_{w}\left(x\right)\prod _{i=1}^{s}{B}_{k,{n}_{i}}^{\left(\alpha \right)}\left(x,q\right)d{\mu }_{-q}\left(x\right)\phantom{\rule{2em}{0ex}}\\ =\prod _{i=1}^{s}\left(\begin{array}{c}\hfill {n}_{i}\hfill \\ \hfill k\hfill \end{array}\right)\underset{{ℤ}_{p}}{\int }{q}^{x\left(h-1\right)}{\varphi }_{w}\left(x\right){\left[x\right]}_{{q}^{\alpha }}^{sk}{\left[1-x\right]}_{{q}^{-\alpha }}^{{n}_{1}+{n}_{2}+\cdots +{n}_{s}-sk}d{\mu }_{-q}\left(x\right)\phantom{\rule{2em}{0ex}}\\ =\prod _{i=1}^{s}\left(\begin{array}{c}\hfill {n}_{i}\hfill \\ \hfill k\hfill \end{array}\right)\underset{{ℤ}_{p}}{\int }{q}^{x\left(h-1\right)}{\varphi }_{w}\left(x\right){\left(1-{\left[1-x\right]}_{{q}^{-\alpha }}\right)}^{sk}{\left[1-x\right]}_{{q}^{-\alpha }}^{{n}_{1}+{n}_{2}+\cdots +{n}_{s}-sk}d{\mu }_{-q}\left(x\right)\phantom{\rule{2em}{0ex}}\\ =\prod _{i=1}^{s}\left(\begin{array}{c}\hfill {n}_{i}\hfill \\ \hfill k\hfill \end{array}\right)\sum _{l=0}^{sk}\left(\begin{array}{c}\hfill sk\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{sk-l}\underset{{ℤ}_{p}}{\int }{q}^{x\left(h-1\right)}{\varphi }_{w}\left(x\right){\left[1-x\right]}_{{q}^{-\alpha }}^{{n}_{1}+{n}_{2}+\cdots +{n}_{s}-l}d{\mu }_{-q}\left(x\right)\phantom{\rule{2em}{0ex}}\\ =\prod _{i=1}^{s}\left(\begin{array}{c}\hfill {n}_{i}\hfill \\ \hfill k\hfill \end{array}\right)\sum _{l=0}^{sk}\left(\begin{array}{c}\hfill sk\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{sk-l}\left({\left[2\right]}_{q}+w{q}^{1+h}\frac{{G}_{{n}_{1}+{n}_{2}+\cdots +{n}_{s}-l+1,{q}^{-1},{w}^{-1}}^{\left(h,\alpha \right)}}{{n}_{1}+{n}_{2}+\cdots +{n}_{s}-l+1}\right).\phantom{\rule{2em}{0ex}}\end{array}$

Therefore, we have the theorem below.

Theorem 10. For ${n}_{1},{n}_{2},{n}_{3},\dots ,{n}_{s},k\in {ℤ}_{+}$ with n1 + n2 + . . . + n s > sk + 1, we have

$\begin{array}{c}\underset{{ℤ}_{p}}{\int }{q}^{x\left(h-1\right)}{\varphi }_{w}\left(x\right)\prod _{i=1}^{s}{B}_{k,{n}_{i}}^{\left(\alpha \right)}\left(x,q\right)d{\mu }_{-q}\left(x\right)\\ =\prod _{i=1}^{s}\left(\begin{array}{c}\hfill {n}_{i}\hfill \\ \hfill k\hfill \end{array}\right)\sum _{l=0}^{sk}\left(\begin{array}{c}\hfill sk\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{sk-l}\left({\left[2\right]}_{q}+w{q}^{1+h}\frac{{G}_{{n}_{1}+{n}_{2}+\cdots +{n}_{s}-l+1,{q}^{-1},{w}^{-1}}^{\left(h,\alpha \right)}}{{n}_{1}+{n}_{2}+\cdots +{n}_{s}-l+1}\right).\end{array}$

In the same manner as in (2.11), we can get the following relation:

$\begin{array}{c}\underset{{ℤ}_{p}}{\int }{q}^{x\left(h-1\right)}{\varphi }_{w}\left(x\right)\prod _{i=1}^{s}{B}_{k,{n}_{i}}^{\left(\alpha \right)}\left(x,q\right)d{\mu }_{-q}\left(x\right)\\ =\prod _{i=1}^{s}\left(\begin{array}{c}\hfill {n}_{i}\hfill \\ \hfill k\hfill \end{array}\right)\underset{{ℤ}_{p}}{\int }{q}^{x\left(h-1\right)}{\varphi }_{w}\left(x\right){\left[x\right]}_{{q}^{\alpha }}^{sk}{\left(1-{\left[x\right]}_{{q}^{\alpha }}\right)}^{{n}_{1}+{n}_{2}+\cdots +{n}_{s}-sk}d{\mu }_{-q}\left(x\right)\\ =\prod _{i=1}^{s}\left(\begin{array}{c}\hfill {n}_{i}\hfill \\ \hfill k\hfill \end{array}\right)\underset{{ℤ}_{p}}{\int }{q}^{x\left(h-1\right)}{\varphi }_{w}\left(x\right){\left[x\right]}_{{q}^{\alpha }}^{sk}\sum _{l=0}^{{n}_{1}+{n}_{2}+\cdots +{n}_{s}-sk}{\left(-1\right)}^{l}\left(\begin{array}{c}{n}_{1}+{n}_{2}+\cdots +{n}_{s}-sk\\ l\end{array}\right){\left(-1\right)}^{l}{\left[x\right]}_{{q}^{\alpha }}^{l}d{\mu }_{-q}\left(x\right)\\ =\prod _{i=1}^{s}\left(\begin{array}{c}\hfill {n}_{i}\hfill \\ \hfill k\hfill \end{array}\right)\sum _{l=0}^{{n}_{1}+{n}_{2}+\cdots +{n}_{s}-sk}{\left(-1\right)}^{l}\left(\begin{array}{c}{n}_{1}+{n}_{2}+\cdots +{n}_{s}-sk\\ l\end{array}\right)\underset{{ℤ}_{p}}{\int }{q}^{x\left(h-1\right)}{\varphi }_{w}\left(x\right){\left[x\right]}_{{q}^{\alpha }}^{sk+l}d{\mu }_{-q}\left(x\right)\\ =\prod _{i=1}^{s}\left(\begin{array}{c}\hfill {n}_{i}\hfill \\ \hfill k\hfill \end{array}\right)\sum _{l=0}^{{n}_{1}+{n}_{2}+\cdots +{n}_{s}-sk}{\left(-1\right)}^{l}\left(\begin{array}{c}{n}_{1}+{n}_{2}+\cdots +{n}_{s}-sk\\ l\end{array}\right)\frac{{G}_{sk+l+1,q,w}^{\left(h,\alpha \right)}}{sk+l+1},\end{array}$

where ${n}_{1},{n}_{2},\dots ,{n}_{s},k\in {ℤ}_{+}$ with n1 + n2 + . . . + n s > sk + 1.

By Theorem 11 and (2.9), we have the following corollary.

Corollary 11. Let $m\in ℕ$. For ${n}_{1},{n}_{2},\dots ,{n}_{s},k\in {ℤ}_{+}$ with n1 + . . . + n s > sk + 1, we have

$\begin{array}{l}\sum _{l=0}^{sk}\left(\begin{array}{c}\hfill sk\hfill \\ \hfill l\hfill \end{array}\right){\left(-1\right)}^{s{k}^{-l}}\left({\left[2\right]}_{q}+w{q}^{1+h}\frac{{G}_{{n}_{1}+{n}_{2}+\cdots +{n}_{s}-l+1,{q}^{-1},{w}^{-1}}^{\left(h,\alpha \right)}}{{n}_{1}+{n}_{2}+\cdots +{n}_{s}-l+1}\right)\phantom{\rule{2em}{0ex}}\\ =\sum _{l=0}^{{n}_{1}+{n}_{2}+\cdots +{n}_{s}-sk}{\left(-1\right)}^{l}\left(\begin{array}{c}{n}_{1}+{n}_{2}+\cdots +{n}_{s}-sk\\ l\end{array}\right)\frac{{G}_{sk+l+1,q,w}^{\left(h,\alpha \right)}}{sk+l+1},\phantom{\rule{2em}{0ex}}\end{array}$

## References

1. Jung NS, Ryoo CS: On the twisted (h; q)-Genocchi numbers and polynomials associated with weight α. Proc. Jangjeon Math. Soc. 2012, 15: 1–9. 10.1155/2011/123483

2. Rim SH, Jin JH, Moon EJ, Lee SJ: Some identities on the q -Genocchi polynomials of higher-order and q -Stirling numbers by the fermionic p -adic integral on ${ℤ}_{p}$ . International Journal of Mathematics and Mathematical Sciences 2010., 2010: Art. ID 860280

3. Ryoo CS: Some identities of the twisted q -Euler numbers and polynomials associated with q -Bernstein polynomials. Proc Jangjeon Math Soc 2011, 14: 239–248.

4. Ryoo CS: Some relations between twisted q -Euler numbers and Bernstein polynomials. Adv Stud Contemp Math 2011, 21: 217–223.

5. Simesk Y, Kurt V, Kim D: New approach to the complete sum of products of the twisted ( h, q )-Bernoulli numbers and polynomials. J Nonlinear Math Phys 2007, 14: 44–56. 10.2991/jnmp.2007.14.1.5

6. Kim T, Choi J, Kim Y-H: Some identities on the q -Bernstein polynomials, q -Stirling numbers and q -Bernoulli numbers. Adv Stud Contemp Math 2010, 20: 335–341.

7. Kim T, Jang LC, Yi H: A note on the modified q -Bernstein polynomials. Discrete Dynamics in Nature and Society 2010, 2010: 12. Article ID 706483

8. Kim T: Some identities on the q -Euler polynomials of higher order and q -Stirling numbers by the fermionic p -adic integral on ${ℤ}_{p}$ . Russian J Math Phys 2009, 16: 484–491. 10.1134/S1061920809040037

9. Kim T: Note on the Euler numbers and polynomials. Adv Stud Contemp Math 2008, 17: 131–136.

10. Kim T: q -Volkenborn integration. Russ J Math Phys 2002, 9: 288–299.

11. Kim T, Choi J, Kim YH, Ryoo CS: On the fermionic p -adic integral representation of Bernstein polynomials associated with Euler numbers and polynomials. J Inequal Appl 2010, 2010: 12. Art ID 864247

12. Simsek Y, Acikgoz M: A new generating function of q -Bernstein-type polynomials and their interpolation function. Abstract and Applied Analysis 2010, 2010: 12. Article ID 769095

## Author information

Authors

### Corresponding author

Correspondence to Nam-Soon Jung.

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

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

## Rights and permissions

Reprints and Permissions

Jung, NS., Lee, HY., Kang, JY. et al. The twisted (h, q)-Genocchi numbers and polynomials with weight α and q-bernstein polynomials with weight α. J Inequal Appl 2012, 67 (2012). https://doi.org/10.1186/1029-242X-2012-67 