1 Introduction

The q-calculus theory is a novel theory that is based on finite difference re-scaling. First results in q-calculus belong to Euler, who discovered Euler’s identities for q-exponential functions, and Gauss, who discovered q-binomial formula. The systematic development of q-calculus begins from FH Jackson who 1908 reintroduced the Euler-Jackson q-difference operator (Jackson, 1908). One of the important branches of q-calculus is q-special orthogonal polynomials. Also p-adic numbers were invented by Kurt Hensel around the end of the nineteenth century, and these two branches of number theory joined in the link of p-adic integral and developed. In spite of them being already one hundred years old, these special numbers and polynomials, for instance, q-Bernstein polynomials, q-Genocchi numbers and polynomials, etc., are still today enveloped in an aura of mystery within the scientific community. The p-adic integral was used in mathematical physics, for instance, the functional equation of the q-zeta function, q-Stirling numbers and q-Mahler theory of integration with respect to the ring \(\mathbb{Z} _{p}\) together with Iwasawa’s p-adic L functions. During the last ten years, the q-Bernstein polynomials and q-Genocchi polynomials have attracted a lot of interest and have been studied from different points of view along with some generalizations and modifications by a number of researchers. By using the p-adic invariant q-integral on \(\mathbb{Z} _{p}\), Kim [1] constructed p-adic Bernoulli numbers and polynomials with weight α. He also gave the identities on the q-integral representation of the product of several q-Bernstein polynomials and constructed a link between q-Bernoulli polynomials and q-umbral calculus (cf. [2, 3]). Our aim of this paper is also to show that a fermionic p-adic q-integral representation of products of weighted q-Bernstein polynomials of different degrees \(n_{1},n_{2},\ldots \) on \(\mathbb{Z} _{p}\) can be written in terms of q-Genocchi numbers with weight α and β.

Suppose that p is chosen as an odd prime number. Throughout this paper, we make use of the following notations: \(\mathbb{Z} _{p}\) denotes the ring of p-adic rational integers, ℚ denotes the field of rational numbers, \(\mathbb{Q} _{p}\) denotes the field of p-adic rational numbers and \(\mathbb{C} _{p}\) denotes the completion of algebraic closure of \(\mathbb{Q} _{p}\). Let ℕ be the set of natural numbers and \(\mathbb{N} ^{\ast}=\mathbb{N} \cup \{ 0 \} \). The normalized p-adic absolute value is defined by \(\vert p\vert _{p}=\frac{1}{p}\). When one mentions q-extension, q can be variously considered as an indeterminate, a complex number \(q\in \mathbb{C} \), or a p-adic number \(q\in \mathbb{C} _{p}\). If \(q\in \mathbb{C} \), we assume \(\vert q\vert <1\). If \(q\in \mathbb{C} _{p}\), we assume \(\vert q-1\vert _{p}< p^{-\frac{1}{p-1}}\).

Suppose \(UD ( \mathbb{Z} _{p} ) \) is the space of uniformly differentiable functions on \(\mathbb{Z} _{p}\). For \(f\in UD ( \mathbb{Z} _{p} ) \), the fermionic p-adic q-integral on \(\mathbb{Z} _{p}\) is defined by Kim (see [4, 5]):

$$ I_{-q} ( f ) =\int_{\mathbb{Z} _{p}}f ( \xi )\,d\mu_{-q} ( \xi ) =\lim_{N\rightarrow \infty}\frac{1}{ [ p^{N} ] _{-q}}\sum _{\xi =0}^{p^{N}-1}q^{\xi }f ( \xi ) ( -1 ) ^{\xi}. $$
(1.1)

For \(\alpha,k,n\in \mathbb{N} ^{\ast}\) and \(x\in [ 0,1 ] \), Kim et al. defined weighted q-Bernstein polynomials as follows:

$$ B_{k,n}^{ ( \alpha ) } ( x,q ) =\binom {n}{k} [ x ] _{q^{\alpha}}^{k} [ 1-x ] _{q^{-\alpha }}^{n-k}\quad( \mbox{see [6] and [7]}). $$
(1.2)

If we put \(q\rightarrow1\) and \(\alpha=1\) in Eq. (1.2), since \([ x ] _{q^{\alpha}}^{k}\rightarrow x^{k}\), \([ 1-x ] _{q^{-\alpha}}^{n-k}\rightarrow ( 1-x ) ^{n-k}\), it turns out to be the classical Bernstein polynomials (see [8] and [9]).

The q-extension of x, \([ x ] _{q}\), is defined by

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

Note that \(\lim_{q\rightarrow1} [ x ] _{q}=x\) (for more information, see [1–24]).

In [11], for \(n\in \mathbb{N} ^{\ast}\), modified q-Genocchi numbers with weight α and β are defined by Araci et al. as follows:

$$\begin{aligned} \frac{g_{n+1,q}^{ ( \alpha,\beta ) } ( x ) }{n+1} =&\int_{\mathbb{Z} _{p}}q^{-\beta\xi} [ x+\xi ] _{q^{\alpha}}^{n}\,d\mu _{-q^{\beta }} ( \xi ) \\ =&\frac{ [ 2 ] _{q^{\beta}}}{ [ \alpha ] _{q}^{n} ( 1-q ) ^{n}}\sum_{l=0}^{n} \binom{n}{l} ( -1 ) ^{l}q^{\alpha \ell x}\frac{1}{1+q^{\alpha\ell}} \\ =& [ 2 ] _{q^{\beta}}\sum_{m=0}^{\infty} ( -1 ) ^{m} [ m+x ] _{q^{\alpha}}^{n}. \end{aligned}$$
(1.3)

In the case, for \(x=0\), we have \(g_{n,q}^{ ( \alpha,\beta ) } ( 0 ) =g_{n,q}^{ ( \alpha,\beta ) }\) that are called q-Genocchi numbers with weight α and β.

In [11], for \(\alpha\in \mathbb{N} ^{\ast}\) and \(n\in \mathbb{N} \), q-Genocchi numbers with weight α and β are defined by Araci et al. as follows:

$$ g_{0,q}^{ ( \alpha,\beta ) }=0,\quad\mbox{and}\quad g_{n,q}^{ ( \alpha ,\beta ) } ( 1 ) +g_{n,q}^{ ( \alpha,\beta ) }=\left \{ \begin{array}{@{}l@{\quad}l} [ 2 ] _{q^{\beta}}&\mbox{if }n=1,\\ 0&\mbox{if }n>1. \end{array} \right . $$
(1.4)

In this paper, we obtain some relations between the weighted q-Bernstein polynomials and the modified q-Genocchi numbers with weight α and β. From these relations, we derive some interesting identities on the q-Genocchi numbers with weight α and β.

2 On the weighted q-Genocchi numbers and polynomials

In this part, we will give some arithmetical properties of weighted q-Genocchi polynomials by using the techniques of p-adic integral and the method of generating functions. Thus, by utilizing the definition of weighted q-Genocchi polynomials, we have

$$\begin{aligned} \frac{g_{n+1,q}^{ ( \alpha,\beta ) } ( x ) }{n+1} =&\int_{\mathbb{Z} _{p}}q^{-\beta\xi} [ x+\xi ] _{q^{\alpha}}^{n}\,d\mu _{-q^{\beta }} ( \xi ) \\ =&\int_{\mathbb{Z} _{p}}q^{-\beta\xi} \bigl( [ x ] _{q^{\alpha}}+q^{\alpha x} [ \xi ] _{q^{\alpha}} \bigr) ^{n}\,d\mu_{-q} ( \xi ) \\ =&\sum_{k=0}^{n}\binom{n}{k} [ x ] _{q^{\alpha }}^{n-k}q^{\alpha kx}\int_{\mathbb{Z} _{p}}q^{-\beta\xi} [ \xi ] _{q^{\alpha}}^{k}\,d\mu _{-q} ( \xi ) \\ =&\sum_{k=0}^{n}\binom{n}{k} [ x ] _{q^{\alpha }}^{n-k}q^{\alpha kx}\frac{g_{k+1,q}^{ ( \alpha,\beta ) }}{k+1}. \end{aligned}$$

Thus we state the following theorem.

Theorem 1

Suppose \(n,\alpha,\beta\in \mathbb{N} ^{\ast}\). Then we have

$$ g_{n,q}^{ ( \alpha,\beta ) } ( x ) =q^{-\alpha x}\sum _{k=0}^{n}\binom{n}{k}q^{\alpha kx}g_{k,q}^{ ( \alpha ,\beta ) } [ x ] _{q^{\alpha}}^{n-k}. $$
(2.1)

Moreover,

$$ g_{n,q}^{ ( \alpha,\beta ) } ( x ) =q^{-\alpha x} \bigl( q^{\alpha x}g_{q}^{ ( \alpha,\beta ) }+ [ x ] _{q^{\alpha}} \bigr) ^{n}, $$
(2.2)

by using the umbral (symbolic) convention \(( g_{q}^{ ( \alpha ,\beta ) } ) ^{n}=g_{n,q}^{ ( \alpha,\beta ) }\).

By expression of (1.3), we get

$$\begin{aligned} \frac{g_{n+1,q^{-1}}^{ ( \alpha,\beta ) } ( 1-x ) }{n+1} =&\int_{\mathbb{Z} _{p}}q^{\beta\xi} [ 1-x+\xi ] _{q^{-\alpha}}^{n}\,d\mu _{-q^{-\beta}} ( \xi ) \\ =&\frac{ [ 2 ] _{q^{-\beta}}}{ ( 1-q^{-\alpha } ) ^{n}}\sum_{l=0}^{n} \binom{n}{l} ( -1 ) ^{l}q^{-\alpha\ell ( 1-x ) }\frac{1}{1+q^{-\alpha\ell}} \\ =& ( -1 ) ^{n}q^{\alpha n-\beta} \Biggl( \frac{ [ 2 ] _{q^{\beta}}}{ ( 1-q^{\alpha} ) ^{n}}\sum _{l=0}^{n}\binom {n}{l} ( -1 ) ^{l}q^{\alpha lx}\frac{1}{1+q^{\alpha l}} \Biggr) \\ =& ( -1 ) ^{n}q^{\alpha n-\beta}\frac{g_{n+1,q}^{ ( \alpha ,\beta ) } ( x ) }{n+1}. \end{aligned}$$

Consequently, we obtain the following theorem.

Theorem 2

The following

$$ g_{n+1,q^{-1}}^{ ( \alpha,\beta ) } ( 1-x ) = ( -1 ) ^{n}q^{\alpha n-\beta}g_{n+1,q}^{ ( \alpha,\beta ) } ( x ) $$
(2.3)

is true.

From expression of (2.2) and Theorem 1, we get the following theorem.

Theorem 3

The following identity holds:

$$ g_{0,q}^{ ( \alpha,\beta ) }=0,\quad\textit{and}\quad q^{-\alpha } \bigl( q^{\alpha}g_{q}^{ ( \alpha,\beta ) }+1 \bigr) ^{n}+g_{n,q}^{ ( \alpha,\beta ) }= \left \{ \begin{array}{@{}l@{\quad}l} [2 ]_{q^{\beta}}&\textit{if }n=1,\\ 0&\textit{if }n>1, \end{array} \right . $$

with the usual convention about replacing \(( g_{q}^{ ( \alpha ,\beta ) } ) ^{n}\) by \(g_{n,q}^{ ( \alpha,\beta ) }\).

For \(n,\alpha\in \mathbb{N} \), by Theorem 3, we note that

$$\begin{aligned} q^{2\alpha}g_{n,q}^{ ( \alpha,\beta ) } ( 2 ) =& \bigl( q^{\alpha} \bigl( q^{\alpha}g_{q}^{ ( \alpha,\beta ) }+1 \bigr) +1 \bigr) ^{n} \\ =&\sum_{k=0}^{n}\binom{n}{k}q^{k\alpha} \bigl( q^{\alpha }g_{q}^{ ( \alpha,\beta ) }+1 \bigr) ^{k} \\ =& \bigl( q^{\alpha}g_{q}^{ ( \alpha,\beta ) }+1 \bigr) ^{0}+nq^{\alpha} \bigl( q^{\alpha}g_{q}^{ ( \alpha,\beta ) }+1 \bigr) ^{1} +\sum_{k=2}^{n}\binom{n}{k}q^{k\alpha} \bigl( q^{\alpha }g_{q}^{ ( \alpha,\beta ) }+1 \bigr) ^{k} \\ =& n q^{2\alpha} [ 2 ] _{q^{\beta}}-q^{\alpha}\sum _{k=0}^{n}\binom{n}{k}q^{\alpha k}g_{k,q}^{ ( \alpha,\beta ) } \\ =& n q^{2 \alpha} [ 2 ] _{q^{\beta}}+q^{\alpha }g_{n,q}^{ ( \alpha,\beta ) } \quad\mbox{if }n>1. \end{aligned}$$

Consequently, we state the following theorem.

Theorem 4

Suppose \(n\in \mathbb{N} \). Then we have

$$ g_{n,q}^{ ( \alpha,\beta ) } ( 2 ) ={ n [ 2 ] _{q^{\beta}}}+ \dfrac{g_{n,q}^{ ( \alpha,\beta ) }}{q^{\alpha}}. $$

From expression of Theorem 2 and (2.3), we easily see that

$$ \begin{aligned}[b] & ( n+1 ) q^{-\beta}\int_{\mathbb{Z} _{p}}q^{-\beta\xi} [ 1-\xi ] _{q^{-\alpha}}^{n}\,d\mu _{-q^{\beta }} ( \xi )\\ &\quad= ( -1 ) ^{n}q^{n\alpha-\beta}\int_{\mathbb{Z} _{p}}q^{-\beta\xi} [ \xi-1 ] _{q^{\alpha}}^{n}\,d\mu _{-q^{\beta }} ( \xi ) \\ &\quad= ( -1 ) ^{n}q^{n\alpha-\beta}g_{n+1,q}^{ ( \alpha ,\beta ) } ( -1 ) =g_{n+1,q^{-1}}^{ ( \alpha,\beta ) } ( 2 ) . \end{aligned} $$
(2.4)

Thus, we obtain the following theorem.

Theorem 5

The following identity

$$ ( n+1 ) q^{-\beta}\int_{\mathbb{Z} _{p}}q^{-\beta\xi} [ 1-\xi ] _{q^{-\alpha}}^{n}\,d\mu _{-q^{\beta }} ( \xi ) =g_{n+1,q^{-1}}^{ ( \alpha,\beta ) } ( 2 ) $$

is true.

Suppose \(n,\alpha\in \mathbb{N} \). By expression of Theorem 4 and Theorem 5, we get

$$\begin{aligned} & ( n+1 ) q^{-\beta}\int_{\mathbb{Z} _{p}}q^{-\beta\xi} [ 1-\xi ] _{q^{-\alpha}}^{n}\,d\mu _{-q^{\beta }} ( \xi ) \\ &\quad= ( {n+1} ) q^{-\beta}{ [ 2 ] _{q^{\beta }}}+q^{\alpha }g_{n+1,q^{-1}}^{ ( \alpha,\beta ) }. \end{aligned}$$
(2.5)

For (2.5), we obtain the corollary as follows.

Corollary 1

Suppose \(n,\alpha\in \mathbb{N} ^{\ast}\). Then we have

$$ \int_{\mathbb{Z} _{p}}q^{-\beta\xi} [ 1-\xi ] _{q^{-\alpha}}^{n}\,d\mu _{-q^{\beta }} ( \xi ) = [ 2 ] _{q^{\beta}}+q^{\alpha -\beta} \frac{g_{n+1,q^{-1}}^{ ( \alpha,\beta ) }}{n+1}. $$

3 Novel identities on the weighted q-Genocchi numbers

In this section, we develop modified q-Genocchi numbers with weight α and β, namely we derive interesting and worthwhile relations in analytic number theory.

For \(x\in \mathbb{Z} _{p}\), the p-adic analogues of weighted q-Bernstein polynomials are given by

$$ B_{k,n}^{ ( \alpha ) } ( x,q ) =\binom {n}{k} [ x ] _{q^{\alpha}}^{k} [ 1-x ] _{q^{-\alpha }}^{n-k},\quad \mbox{where }n,k,\alpha\in \mathbb{N} ^{\ast}. $$
(3.1)

By expression of (3.1), Kim et al. get the symmetry of q-Bernstein polynomials weight α as follows:

$$ B_{k,n}^{ ( \alpha ) } ( x,q ) =B_{n-k,n}^{ ( \alpha ) } \bigl( 1-x,q^{-1} \bigr) \quad(\mbox{for details, see [7]}). $$
(3.2)

Thus, from Corollary 1, (3.1) and (3.2), we see that

$$\begin{aligned} &\int_{\mathbb{Z} _{p}}B_{k,n}^{ ( \alpha ) } ( \xi,q ) q^{-\beta \xi }\,d\mu_{-q^{\beta}} ( \xi )\\ &\quad=\int_{\mathbb{Z} _{p}}B_{n-k,n}^{ ( \alpha ) } \bigl( 1-\xi,q^{-1} \bigr) q^{-\beta \xi}\,d\mu_{-q^{\beta}} ( \xi ) \\ &\quad=\binom{n}{k}\sum_{l=0}^{k} \binom{k}{l} ( -1 ) ^{k+l}\int_{\mathbb{Z} _{p}}q^{-\beta\xi} [ 1-\xi ] _{q^{-\alpha}}^{n-l}\,d\mu _{-q^{\beta}} ( \xi ) \\ &\quad=\binom{n}{k}\sum_{l=0}^{k} \binom{k}{l} ( -1 ) ^{k+l} \biggl( [ 2 ] _{q^{\beta}}+q^{\alpha-\beta} \frac {g_{n-l+1,q^{-1}}^{ ( \alpha,\beta ) }}{n-l+1} \biggr) . \end{aligned}$$

For \(n, k\in \mathbb{N} ^{\ast}\) and \(\alpha\in \mathbb{N} \) with \(n>k\), we obtain

$$\begin{aligned} &\int_{\mathbb{Z} _{p}}B_{k,n}^{ ( \alpha ) } ( \xi,q ) q^{-\beta \xi }\,d\mu_{-q^{\beta}} ( \xi ) \\ &\quad=\binom{n}{k}\sum_{l=0}^{k} \binom{k}{l} ( -1 ) ^{k+l} \biggl( [ 2 ] _{q^{\beta}}+q^{\alpha-\beta} \frac {g_{n-l+1,q^{-1}}^{ ( \alpha,\beta ) }}{n-l+1} \biggr) \\ &\quad=\left \{ \begin{array}{@{}l@{\quad}l} [ 2 ] _{q^{\beta}}+q^{\alpha-\beta}\frac {g_{n+1,q^{-1}}^{ ( \alpha,\beta ) }}{n+1} & \mbox{if }k=0, \\ \binom{n}{k}\sum_{l=0}^{k}\binom{k}{l} ( -1 ) ^{k+l} ( [ 2 ] _{q^{\beta}}+q^{\alpha-\beta}\frac{g_{n-l+1,q^{-1}}^{ ( \alpha,\beta ) }}{n-l+1} ) & \mbox{if }k>0.\end{array} \right . \end{aligned}$$
(3.3)

Let us take the fermionic p-adic q-integral on \(\mathbb{Z} _{p}\) on the weighted q-Bernstein polynomials of degree n as follows:

$$\begin{aligned} &\int_{\mathbb{Z} _{p}}B_{k,n}^{ ( \alpha ) } ( \xi,q ) q^{-\beta \xi }\,d\mu_{-q^{\beta}} ( \xi ) \\ &\quad=\binom{n}{k}\int _{\mathbb{Z} _{p}}q^{-\beta\xi} [ \xi ] _{q^{\alpha}}^{k} [ 1-\xi ] _{q^{-\alpha}}^{n-k}\,d\mu_{-q^{\beta}} ( \xi ) \\ &\quad=\binom{n}{k}\sum_{l=0}^{n-k} \binom{n-k}{l} ( -1 ) ^{l}\frac{g_{l+k+1,q}^{ ( \alpha,\beta ) }}{l+k+1}. \end{aligned}$$
(3.4)

Consequently, by expression of (3.3) and (3.4), we state the following theorem.

Theorem 6

The following identity holds:

$$\begin{aligned} &\sum_{l=0}^{n-k}\binom{n-k}{l} ( -1 ) ^{l}\frac {g_{l+k+1,q}^{ ( \alpha,\beta ) }}{l+k+1}\\ &\quad=\left \{ \begin{array}{@{}l@{\quad}l} [ 2 ] _{q^{\beta}}+q^{\alpha-\beta}\frac {g_{n+1,q^{-1}}^{ ( \alpha,\beta ) }}{n+1} & \textit{if }k=0, \\ \sum_{l=0}^{k}\binom{k}{l} ( -1 ) ^{k+l} ( [ 2 ] _{q^{\beta}}+q^{\alpha-\beta}\frac{g_{n-l+1,q^{-1}}^{ ( \alpha ,\beta ) }}{n-l+1} ) & \textit{if }k>0.\end{array} \right . \end{aligned}$$

Suppose \(n_{1},n_{2},k\in \mathbb{N} ^{\ast}\) and \(\alpha\in \mathbb{N} \) with \(n_{1}+n_{2}>2k\). It yields

$$\begin{aligned} &\int_{\mathbb{Z} _{p}}B_{k,n_{1}}^{ ( \alpha ) } ( \xi,q ) B_{k,n_{2}}^{ ( \alpha ) } ( \xi,q ) q^{-\beta \xi}\,d\mu _{-q^{\beta}} ( \xi ) \\ &\quad=\binom{n_{1}}{k}\binom{n_{2}}{k}\sum_{l=0}^{2k} \binom {2k}{l} ( -1 ) ^{2k+l}\int_{\mathbb{Z} _{p}}q^{-\beta\xi} [ 1-\xi ] _{q^{-\alpha }}^{n_{1}+n_{2}-l}\,d\mu _{-q^{\beta}} ( \xi ) \\ &\quad= \Biggl( \binom{n_{1}}{k}\binom{n_{2}}{k}\sum _{l=0}^{2k}\binom {2k}{l} ( -1 ) ^{2k+l} \biggl( [ 2 ] _{q^{\beta}}+q^{\alpha -\beta}\frac{g_{n_{1}+n_{2}-l+1,q^{-1}}^{ ( \alpha,\beta ) }}{n_{1}+n_{2}-l+1} \biggr) \Biggr) \\ &\quad=\left \{ \begin{array}{@{}l@{\quad}l} [ 2 ] _{q^{\beta}}+q^{\alpha-\beta}\frac{g_{n_{1}+n_{2}+1,q^{-1}}^{ ( \alpha,\beta ) }}{n_{1}+n_{2}+1} & \mbox{if }k=0, \\ \binom{n_{1}}{k}\binom{n_{2}}{k}\sum_{l=0}^{2k}\binom{2k}{l} ( -1 ) ^{2k+l} ( [ 2 ] _{q^{\beta}}+q^{\alpha-\beta}\frac{ g_{n_{1}+n_{2}-l+1,q^{-1}}^{ ( \alpha,\beta ) }}{n_{1}+n_{2}-l+1} ) & \mbox{if }k\neq0.\end{array} \right . \end{aligned}$$

Therefore, we obtain the following theorem.

Theorem 7

Suppose \(n_{1},n_{2},k\in \mathbb{N} ^{\ast}\) and \(\alpha,\beta\in \mathbb{N} \) with \(n_{1}+n_{2}>2k\), then we have

$$\begin{aligned} &\int_{\mathbb{Z} _{p}}q^{-\beta\xi}B_{k,n_{1}}^{ ( \alpha ) } ( \xi ,q ) B_{k,n_{2}}^{ ( \alpha ) } ( \xi,q )\,d\mu _{-q^{\beta }} ( \xi ) \\ &\quad=\left \{ \begin{array}{@{}l@{\quad}l} [ 2 ] _{q^{\beta}}+q^{\alpha-\beta}\frac{g_{n_{1}+n_{2}+1,q^{-1}}^{ ( \alpha,\beta ) }}{n_{1}+n_{2}+1} & \textit{if }k=0, \\ \binom{n_{1}}{k}\binom{n_{2}}{k}\sum_{l=0}^{2k}\binom{2k}{l} ( -1 ) ^{2k+l} ( [ 2 ] _{q^{\beta}}+q^{\alpha-\beta}\frac{ g_{n_{1}+n_{2}-l+1,q^{-1}}^{ ( \alpha,\beta ) }}{n_{1}+n_{2}-l+1} ) & \textit{if }k\neq0.\end{array} \right . \end{aligned}$$

By using the binomial theorem, we can derive the following equation:

$$\begin{aligned} &\int_{\mathbb{Z} _{p}}B_{k,n_{1}}^{ ( \alpha ) } ( \xi,q ) B_{k,n_{2}}^{ ( \alpha ) } ( \xi,q ) q^{-\beta \xi}\,d\mu _{-q^{\beta}} ( \xi ) \\ &\quad=\prod _{i=1}^{2}\binom{n_{i}}{k}\sum _{l=0}^{n_{1}+n_{2}-2k}\binom{n_{1}+n_{2}-2k}{l} ( -1 ) ^{l}\int_{\mathbb{Z} _{p}} [ \xi ] _{q^{\alpha}}^{2k+l}q^{-\beta\xi}\,d\mu _{-q^{\beta }} ( \xi ) \\ &\quad=\prod _{i=1}^{2}\binom{n_{i}}{k}\sum _{l=0}^{n_{1}+n_{2}-2k}\binom{n_{1}+n_{2}-2k}{l} ( -1 ) ^{l}\frac{g_{l+2k+1,q}^{ ( \alpha ,\beta ) }}{l+2k+1}. \end{aligned}$$
(3.5)

Thus, we can obtain the following corollary.

Corollary 2

Suppose \(n_{1},n_{2},k\in \mathbb{N} ^{\ast}\) and \(\alpha\in \mathbb{N} \) with \(n_{1}+n_{2}>2k\). Then we have

$$\begin{aligned} &\sum_{l=0}^{n_{1}+n_{2}-2k}\binom{n_{1}+n_{2}-2k}{l} ( -1 ) ^{l}\frac{g_{l+2k+1,q}^{ ( \alpha,\beta ) }}{l+2k+1} \\ &\quad=\left \{ \begin{array}{@{}l@{\quad}l} [ 2 ] _{q^{\beta}}+q^{\alpha-\beta}\frac{g_{n_{1}+n_{2}+1,q^{-1}}^{ ( \alpha,\beta ) }}{n_{1}+n_{2}+1} & \textit{if }k=0, \\ \sum_{l=0}^{2k}\binom{2k}{l} ( -1 ) ^{2k+l} ( [ 2 ] _{q^{\beta}}+q^{\alpha-\beta}\frac {g_{n_{1}+n_{2}-l+1,q^{-1}}^{ ( \alpha,\beta ) }}{n_{1}+n_{2}-l+1} ) & \textit{if }k\neq 0. \end{array} \right . \end{aligned}$$

For \(\xi\in \mathbb{Z} _{p}\) and \(s\in \mathbb{N} \) with \(s\geq2\), let \(n_{1},n_{2},\ldots,n_{s},k\in \mathbb{N} ^{\ast}\) and \(\alpha\in \mathbb{N} \) with \(\sum_{l=1}^{s}n_{l}>sk\). Then we take the fermionic p-adic q-integral on \(\mathbb{Z} _{p}\) for the weighted q-Bernstein polynomials of degree n as follows:

$$\begin{aligned} &\int_{\mathbb{Z} _{p}}\underset{s\mbox{-}\mathrm{times}}{ \underbrace{B_{k,n_{1}}^{ ( \alpha ) } ( \xi,q ) B_{k,n_{2}}^{ ( \alpha ) } ( \xi ,q ) \cdots B_{k,n_{s}}^{ ( \alpha ) } ( \xi ,q ) }}q^{-\beta\xi}\,d\mu_{-q^{\beta}} ( \xi ) \\ &\quad=\prod_{i=1}^{s}\binom{n_{i}}{k} \int_{\mathbb{Z} _{p}} [ \xi ] _{q^{\alpha}}^{sk} [ 1-\xi ] _{q^{-\alpha }}^{n_{1}+n_{2}+\cdots+n_{s}-sk}q^{-\beta\xi}\,d\mu_{-q^{\beta}} ( \xi ) \\ &\quad=\prod_{i=1}^{s}\binom{n_{i}}{k} \sum_{l=0}^{sk}\binom {sk}{l} ( -1 ) ^{l+sk}\int_{\mathbb{Z} _{p}}q^{-\beta\xi} [ 1-\xi ] _{q^{-\alpha }}^{n_{1}+n_{2}+\cdots+n_{s}-sk}\,d\mu_{-q^{\beta}} ( \xi ) \\ &\quad=\left \{ \begin{array}{@{}l@{\quad}l} [ 2 ] _{q^{\beta}}+q^{\alpha-\beta}\frac{g_{n_{1}+n_{2}+\cdots+n_{s}+1,q^{-1}}^{ ( \alpha,\beta ) }}{n_{1}+n_{2}+\cdots+n_{s}+1} & \textit{if }k=0, \\ \prod_{i=1}^{s}\binom{n_{i}}{k}\sum_{l=0}^{sk}\binom {sk}{l} ( -1 ) ^{sk+l} ( [ 2 ] _{q^{\beta}}+q^{\alpha -\beta}\frac{g_{n_{1}+n_{2}+\cdots+n_{s}-l+1,q^{-1}}^{ ( \alpha,\beta ) }}{n_{1}+n_{2}+\cdots+n_{s}-l+1} ) & \textit{if }k\neq0. \end{array} \right . \end{aligned}$$

So from above, we have the following theorem.

Theorem 8

Suppose \(s\in \mathbb{N} \) with \(s\geq2\), let \(n_{1},n_{2},\ldots,n_{s},k\in \mathbb{N} ^{\ast}\) and \(\alpha\in \mathbb{N} \) with \(\sum_{l=1}^{s}n_{l}>sk\). Then we have

$$\begin{aligned} &\int_{\mathbb{Z} _{p}}q^{-\beta\xi}\prod _{i=1}^{s}B_{k,n_{i}}^{ ( \alpha ) } ( \xi )\,d\mu_{-q} ( \xi ) \\ &\quad=\left \{ \begin{array}{@{}l@{\quad}l} [ 2 ] _{q^{\beta}}+q^{\alpha-\beta}\frac{g_{n_{1}+n_{2}+\cdots+n_{s}+1,q^{-1}}^{ ( \alpha,\beta ) }}{n_{1}+n_{2}+\cdots+n_{s}+1} & \textit{if }k=0, \\ \prod_{i=1}^{s}\binom{n_{i}}{k}\sum_{l=0}^{sk}\binom {sk}{l} ( -1 ) ^{sk+l} ( [ 2 ] _{q^{\beta}}+q^{\alpha -\beta}\frac{g_{n_{1}+n_{2}+\cdots+n_{s}-l+1,q^{-1}}^{ ( \alpha,\beta ) }}{n_{1}+n_{2}+\cdots+n_{s}-l+1} ) & \textit{if }k\neq0.\end{array} \right . \end{aligned}$$

From the definition of weighted q-Bernstein polynomials and the binomial theorem, we easily get

$$\begin{aligned} &\int_{\mathbb{Z} _{p}}\underset{s\mbox{-}\mathrm{times}}{q^{-\beta\xi}\underbrace{B_{k,n_{1}}^{ ( \alpha ) } ( \xi,q ) B_{k,n_{2}}^{ ( \alpha ) } ( \xi,q ) \cdots B_{k,n_{s}}^{ ( \alpha ) } ( \xi ,q ) }}\,d\mu_{-q^{\beta}} ( \xi ) \\ &\quad=\prod_{i=1}^{s}\binom{n_{i}}{k} \sum_{l=0}^{n_{1}+\cdots+n_{s}-sk}\binom{\sum_{d=1}^{s} ( n_{d}-k ) }{l} ( -1 ) ^{l}\int_{\mathbb{Z} _{p}}q^{-\beta\xi} [ \xi ] _{q^{\alpha}}^{sk+l}\,d\mu _{-q^{\beta }} ( \xi ) \\ &\quad=\prod_{i=1}^{s}\binom{n_{i}}{k} \sum_{l=0}^{n_{1}+\cdots+n_{s}-sk}\binom{\sum_{d=1}^{s} ( n_{d}-k ) }{l} ( -1 ) ^{l}\frac{g_{l+sk+1,q}^{ ( \alpha,\beta ) }}{l+sk+1}. \end{aligned}$$
(3.6)

Therefore, from (3.6) and Theorem 8, we get an interesting corollary as follows.

Corollary 3

Suppose \(s\in \mathbb{N} \) with \(s\geq2\), let \(n_{1},n_{2},\ldots,n_{s},k\in \mathbb{N} ^{\ast}\) and \(\alpha\in \mathbb{N} \) with \(\sum_{l=1}^{s}n_{l}>sk\). Then we have

$$\begin{aligned}& \sum_{l=0}^{n_{1}+\cdots+n_{s}-sk}\binom{\sum_{d=1}^{s} ( n_{d}-k ) }{l} ( -1 ) ^{l}\frac{g_{l+sk+1,q}^{ ( \alpha,\beta ) }}{l+sk+1} \\& \quad=\left \{ \begin{array}{@{}l@{\quad}l} [ 2 ] _{q^{\beta}}+q^{\alpha-\beta}\frac{g_{n_{1}+n_{2}+\cdots+n_{s}+1,q^{-1}}^{ ( \alpha,\beta ) }}{n_{1}+n_{2}+\cdots+n_{s}+1} & \textit{if }k=0, \\ \sum_{l=0}^{sk}\binom{sk}{l} ( -1 ) ^{sk+l} ( [ 2 ] _{q^{\beta}}+q^{\alpha-\beta}\frac {g_{n_{1}+n_{2}+\cdots+n_{s}-l+1,q^{-1}}^{ ( \alpha,\beta ) }}{n_{1}+n_{2}+\cdots+n_{s}-l+1} ) & \textit{if }k\neq0.\end{array} \right . \end{aligned}$$