1 Introduction
Let p be a fixed odd prime number. Throughout this paper, \(\mathbb{Z}_{p}\), \(\mathbb{Q}_{p}\) and \(\mathbb{C}_{p}\), will denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of algebraic closure of \(\mathbb{Q}_{p}\), respectively. Let \(\nu _{p}\) be the normalized exponential valuation of \(\mathbb{C}_{p}\) with \(|p |_{p}=p^{-\nu _{p} (p)}=\frac{1}{p}\). For \(\lambda \in \mathbb{C}_{p}\) with \(|\lambda |_{p} < p^{-\frac{1}{p-1}}\), the degenerate Euler polynomials are defined by the generating function
When \(x=0\), \({\mathcal{E}}_{n,\lambda }={\mathcal{E}}_{n,\lambda }(0)\) are called the degenerate Euler numbers. The degenerate exponential function is defined by
where
$$ \begin{aligned} \frac{2}{(1+\lambda t)^{\frac{1}{\lambda }}+1}(1+\lambda t)^{\frac{x}{ \lambda }}=\sum_{n=0}^{\infty }{ \mathcal{E}}_{n,\lambda }(x)\frac{t ^{n}}{n!} \quad (\text{see [1, 2]}). \end{aligned} $$
(1)
$$ \begin{aligned} e_{\lambda }^{x}(t)=(1+ \lambda t)^{\frac{x}{\lambda }}=\sum_{n=0}^{ \infty }(x)_{n,\lambda } \frac{t^{n}}{n!} \quad (\text{see [6]}), \end{aligned} $$
(2)
$$ \begin{aligned} (x)_{0,\lambda }=1,\qquad (x)_{n,\lambda }=x(x- \lambda ) (x-2\lambda ) \cdots \bigl(x-(n-1)\lambda \bigr),\quad \text{for } n\geq 1. \end{aligned} $$
(3)
From (1), we note that
$$ \begin{aligned} {\mathcal{E}}_{n,\lambda }(x)=\sum _{l=0}^{n}\binom{n}{l}{\mathcal{E}} _{l,\lambda }(x)_{n-l,\lambda }\quad (n\geq 0). \end{aligned} $$
(4)
Anzeige
Recently, Kim–Kim introduced the degenerate Bernstein polynomials given by
Thus, by (5), we get
where n, k are nonnegative integers.
$$ \begin{aligned} \frac{(x)_{k,\lambda }}{k!}t^{k} (1+ \lambda t)^{\frac{1-x}{\lambda }}= \sum_{n=k}^{\infty }B_{k,n}(x| \lambda )\frac{t^{n}}{n!} \quad (\text{see [8--10]}). \end{aligned} $$
(5)
$$ \begin{aligned} B_{k,n}(x|\lambda )=\textstyle\begin{cases} \binom{n}{k}(x)_{k,\lambda }(1-x)_{n-k,\lambda }, & \text{if } n\geq k, \\ 0, & \text{if }n< k, \end{cases}\displaystyle \end{aligned} $$
(6)
Let f be a continuous function on \(\mathbb{Z}_{p}\). Then the degenerate Bernstein operator of order n is given by
The fermionic p-adic integral on \(\mathbb{Z}_{p}\) is defined by Kim as
By (8), we get
From (8), we note that
On the other hand,
By (10) and (11), we get
$$ \begin{aligned}[b] \mathbb{B}_{n,\lambda }(f|\lambda ) &= \sum_{k=0}^{n}f \biggl(\frac{k}{n} \biggr) \binom{n}{k}(x)_{k,\lambda }(1-x)_{n-k,\lambda } \\ &=\sum_{k=0}^{n}f \biggl(\frac{k}{n} \biggr)B_{k,n}(x|\lambda ) \quad (\text{see [8, 9, 12--15, 17--19]}). \end{aligned} $$
(7)
$$ \begin{aligned} \int _{\mathbb{Z}_{p} } f(x)\,d\mu _{-1} (x)= \lim _{N \rightarrow \infty } \sum_{x=0}^{p^{N}-1} f(x) (-1)^{x} \quad (\text{see [3, 9]}). \end{aligned} $$
(8)
$$ \begin{aligned} \int _{\mathbb{Z}_{p} } f(x+1)\,d\mu _{-1} (x)+ \int _{\mathbb{Z}_{p} } f(x)\,d\mu _{-1} (x)=2f(0) \quad ( \text{see [3, 7, 10, 11, 16]}). \end{aligned} $$
(9)
$$ \begin{aligned} \int _{\mathbb{Z}_{p} }(1+\lambda t)^{\frac{x+y}{\lambda }}\,d\mu _{-1}(y)= \frac{2}{(1+ \lambda t)^{\frac{1}{\lambda }}+1}(1+\lambda t)^{\frac{x}{\lambda }}= \sum_{n=0}^{\infty }{ \mathcal{E}}_{n,\lambda }(x)\frac{t^{n}}{n!}. \end{aligned} $$
(10)
$$ \begin{aligned} \int _{\mathbb{Z}_{p} }(1+\lambda t)^{\frac{x+y}{\lambda }}\,d\mu _{-1}(y)= \sum_{n=0}^{\infty } \int _{\mathbb{Z}_{p} }(x+y)_{n,\lambda }\,d\mu _{-1}(y) \frac{t ^{n}}{n!}. \end{aligned} $$
(11)
$$ \begin{aligned} \int _{\mathbb{Z}_{p} }(x+y)_{n,\lambda }\,d\mu _{-1}(y)={ \mathcal{E}}_{n, \lambda }(x) \quad (n\geq 0) \quad (\text{see [8, 9]}). \end{aligned} $$
(12)
The study of degenerate versions of some special polynomials and numbers began with the work of Carlitz on the degenerate Bernoulli and Euler polynomials and numbers in [1, 2]. As a continuation of this initiative of Carlitz, Kim and his colleagues have been introducing various degenerate special polynomials and numbers and investigating their properties, some identities related to them and their applications. This research has been carried out by means of generating functions, combinatorial methods, umbral calculus, p-adic analysis and differential equations (see [8, 9] and the references therein). Here, along the same line and by virtue of fermionic p-adic integrals on \(\mathbb{Z}_{p}\) and generating functions, we investigate some properties and identities for degenerate Euler polynomials related to degenerate Bernstein polynomials. In addition, we study two variable degenerate Bernstein polynomials and the degenerate Bernstein operators.
2 Degenerate Euler and Bernstein polynomials
From (1), we note that
Comparing the coefficients on both sides of (13), we have
where \(\delta _{n,k}\) is the Kronecker symbol.
$$ \begin{aligned}[b] 2 &=\sum _{n=0}^{\infty } \Biggl(\sum_{m=0}^{n} \binom{n}{m}{\mathcal{E}} _{m,\lambda }(1)_{n-m,\lambda }+{ \mathcal{E}}_{n,\lambda } \Biggr)\frac{t ^{n}}{n!} \\ &=\sum_{n=0}^{\infty } \bigl({ \mathcal{E}}_{n,\lambda }(1)+{\mathcal{E}} _{n,\lambda } \bigr) \frac{t^{n}}{n!}. \end{aligned} $$
(13)
$$ \begin{aligned} {\mathcal{E}}_{n,\lambda }(1)+{ \mathcal{E}}_{n,\lambda }=2\delta _{0,n}\quad (n,k\geq 0), \end{aligned} $$
(14)
Anzeige
By (1), we easily get
From (1), (4) and (14), we note that
where n is a positive integer.
$$ \begin{aligned} {\mathcal{E}}_{n,\lambda }(1-x)=(-1)^{n} {\mathcal{E}}_{n,-\lambda }(x) \quad (n\geq 0). \end{aligned} $$
(15)
$$ \begin{aligned}[b] {\mathcal{E}}_{n,\lambda }(2) &= \sum_{l=0}^{n}\binom{n}{l}{\mathcal{E}} _{l,\lambda }(1) (1)_{n-l,\lambda } \\ &=(1)_{n,\lambda }+\sum_{l=1}^{n} \binom{n}{l}{\mathcal{E}}_{l,\lambda }(1) (1)_{n-l,\lambda } \\ &=2(1)_{n,\lambda }-\sum_{l=0}^{n} \binom{n}{l}(1)_{n-l,\lambda } {\mathcal{E}}_{l,\lambda } \\ &=2(1)_{n,\lambda }+{\mathcal{E}}_{n,\lambda }, \end{aligned} $$
(16)
Therefore, by (16), we obtain the following theorem.
Theorem 2.1
For
\(n\in \mathbb{N}\), we have
$$ \begin{aligned} {\mathcal{E}}_{n,\lambda }(2)=2(1)_{n,\lambda }+{ \mathcal{E}}_{n, \lambda }. \end{aligned} $$
Note that
Therefore, by (12), (15) and (17), we easily get
Therefore, by (18) and Theorem 2.1, we obtain the following theorem.
$$ \begin{aligned} (1-x)_{n,\lambda }=(-1)^{n} (x-1)_{n,-\lambda } \quad (n \geq 0). \end{aligned} $$
(17)
$$ \begin{aligned}[b] \int _{\mathbb{Z}_{p} }(1-x)_{n,\lambda }\,d\mu _{-1}(x) &=(-1)^{n} \int _{\mathbb{Z}_{p} }(x-1)_{n,-\lambda }\,d\mu _{-1}(x) \\ &= \int _{\mathbb{Z}_{p} }(x+2)_{n,\lambda }\,d\mu _{-1}(x). \end{aligned} $$
(18)
Theorem 2.2
For
\(n\in \mathbb{N}\), we have
$$ \begin{aligned} \int _{\mathbb{Z}_{p} }(1-x)_{n,\lambda }\,d\mu _{-1}(x)= \int _{\mathbb{Z}_{p} }(x+2)_{n, \lambda }\,d\mu _{-1}(x)=2(1)_{n,\lambda }+ \int _{\mathbb{Z}_{p} }(x)_{n, \lambda }\,d\mu _{-1}(x). \end{aligned} $$
Corollary 2.3
For
\(n\in \mathbb{N}\), we have
$$ \begin{aligned} (-1)^{n} { \mathcal{E}}_{n,-\lambda }(-1)=2(1)_{n,\lambda }+{\mathcal{E}} _{n,\lambda }={\mathcal{E}}_{n,\lambda }(2). \end{aligned} $$
By (4), we get
$$ \begin{aligned} {\mathcal{E}}_{n,\lambda }(1-x) &=\sum _{l=0}^{n}\binom{n}{l}(1-x)_{n-l, \lambda }{ \mathcal{E}}_{l,\lambda } \\ &=\sum_{l=0}^{n}\binom{n}{l}(x)_{l,\lambda }(1-x)_{n-l,\lambda } \frac{ {\mathcal{E}}_{l,\lambda }}{(x)_{l,\lambda }} \\ &=\sum_{l=0}^{n}B_{l,n}(x|\lambda ){\mathcal{E}}_{l,\lambda }\frac{1}{(x)_{l, \lambda }}. \end{aligned} $$
(19)
Let
$$ \begin{aligned} \frac{1}{(x)_{l,\lambda }}=\frac{1}{x(x-\lambda )(x-2\lambda )\cdots (x-(l-1)\lambda )}= \sum_{k=0}^{l-1}\frac{A_{k}}{x-k\lambda }\quad (l \in \mathbb{N}). \end{aligned} $$
(20)
Then we have
By (20) and (21), we get
From (20) and (22), we have
By (19) and (20), we get
Therefore, by (24), we obtain the following theorem.
$$ \begin{aligned} A_{k}=\lambda ^{1-l} \prod^{l-1}_{\substack{i=0,\\i\neq k}} \biggl(\frac{1}{k-i} \biggr)= \lambda ^{1-l}\frac{(-1)^{k-l-1}}{k!(l-1-k)!}= \frac{\lambda ^{1-l}}{(l-1)!} \binom{l-1}{k}(-1)^{k-l-1}. \end{aligned} $$
(21)
$$ \begin{aligned} A_{k}=\frac{(-\lambda )^{1-l}}{(l-1)!} \binom{l-1}{k}(-1)^{k}. \end{aligned} $$
(22)
$$ \begin{aligned} \frac{1}{(x)_{l,\lambda }}=\sum _{k=0}^{l-1}\frac{(-1)^{k}}{(l-1)!} \binom{l-1}{k} \frac{(-\lambda )^{1-l}}{x-k\lambda }\quad (l\in \mathbb{N}). \end{aligned} $$
(23)
$$ \begin{aligned} [b] {\mathcal{E}}_{n,\lambda }(1-x) &=\sum _{l=0}^{n}B_{l,n}(x|\lambda ) { \mathcal{E}}_{l,\lambda }\frac{1}{(x)_{l,\lambda }} \\ &=(1-x)_{n,\lambda }+\sum_{l=1}^{n}B_{l,n}(x| \lambda ){\mathcal{E}} _{l,\lambda }\frac{1}{(x)_{l,\lambda }} \\ &=(1-x)_{n,\lambda }+\sum_{l=1}^{n}B_{l,n}(x| \lambda ){\mathcal{E}} _{l,\lambda }\frac{(-\lambda )^{1-l}}{(l-1)!}\sum _{k=0}^{l-1}(-1)^{k} \binom{l-1}{k} \frac{1}{x-k\lambda }. \end{aligned} $$
(24)
Theorem 2.4
For
\(n\geq 0\), we have
$$ \begin{aligned} {\mathcal{E}}_{n,\lambda }(1-x)=(1-x)_{n,\lambda }+ \sum_{l=1}^{n}B _{l,n}(x|\lambda ){ \mathcal{E}}_{l,\lambda } \frac{(-\lambda )^{1-l}}{(l-1)!}\sum_{k=0}^{l-1}(-1)^{k} \binom{l-1}{k}\frac{1}{x-k\lambda }. \end{aligned} $$
Corollary 2.5
For
\(n\geq 0\), we have
$$ \begin{aligned} {\mathcal{E}}_{n,\lambda }(2)=(2)_{n,\lambda }- \sum_{l=1}^{n}B_{l,n}(-1| \lambda ){\mathcal{E}}_{l,\lambda }\frac{(-\lambda )^{1-l}}{(l-1)!} \sum _{k=0}^{l-1}(-1)^{k}\binom{l-1}{k} \frac{1}{1+k\lambda }. \end{aligned} $$
For \(k\in \mathbb{N}\), the higher order degenerate Euler polynomials are given by the generating function
From (5) and (25), we note that
Therefore, by comparing the coefficients on both sides of (26), we obtain the following theorem.
$$ \begin{aligned} \biggl(\frac{2}{(1+\lambda t)^{\frac{1}{\lambda }}+1} \biggr)^{k}(1+ \lambda t)^{\frac{x}{\lambda }}=\sum _{n=0}^{\infty }{\mathcal{E}}_{n, \lambda }^{(k)}(x) \frac{t^{n}}{n!} \quad (\text{see [4, 5]}). \end{aligned} $$
(25)
$$ \begin{aligned}[b] \sum_{n=0}^{\infty } \frac{1}{\binom{n+k}{n}}B_{k,n+k}(x|\lambda )\frac{t ^{n}}{n!} &=(x)_{k,\lambda }(1+\lambda t)^{\frac{1-x}{\lambda }} \\ &=\frac{(x)_{k,\lambda }}{2^{k}}\sum_{l=0}^{k} \binom{k}{l} \biggl(\frac{2}{(1+ \lambda t)^{\frac{1}{\lambda }}+1} \biggr)^{k}(1+\lambda t)^{\frac{1-x+l}{ \lambda }} \\ &=\frac{(x)_{k,\lambda }}{2^{k}}\sum_{n=0}^{\infty } \Biggl( \sum_{l=0} ^{k}\binom{k}{l}{ \mathcal{E}}_{n,\lambda }^{(k)}(1-x+l) \Biggr)\frac{t ^{n}}{n!}. \end{aligned} $$
(26)
Theorem 2.6
For
\(n,k\in \mathbb{N}\), we have
$$ \begin{aligned} \frac{2^{k}}{\binom{n+k}{n}}B_{k,n+k}(x| \lambda )=(x)_{k,\lambda } \sum_{l=0}^{k} \binom{k}{l}{\mathcal{E}}_{n,\lambda }^{(k)}(1-x+l). \end{aligned} $$
Let f be a continuous function on \(\mathbb{Z}_{p}\). For \(x_{1},x_{2} \in \mathbb{Z}_{p}\), we consider the degenerate Bernstein operator of order n given by
where
where n, k are nonnegative integers.
$$ \begin{aligned} [b] \mathbb{B}_{n,\lambda }(f|x_{1},x_{2}) &=\sum_{k=0}^{n}f \biggl(\frac{k}{n} \biggr) \binom{n}{k}(x_{1})_{k,\lambda }(1-x_{2})_{n-k,\lambda } \\ &=\sum_{k=0}^{n}f \biggl(\frac{k}{n} \biggr)B_{k,n}(x_{1},x_{2}| \lambda ), \end{aligned} $$
(27)
$$ \begin{aligned} B_{k,n}(x_{1},x_{2}| \lambda )=\binom{n}{k}(x_{1})_{k,\lambda }(1-x _{2})_{n-k,\lambda }, \end{aligned} $$
(28)
Here, \(B_{k,n}(x_{1},x_{2}|\lambda )\) are called two variable degenerate Bernstein polynomials of degree n.
From (28), we note that
Thus, by (29), we get
where k is a nonnegative integer. By (28), we easily get
Now, we observe that
Therefore, by (32), we obtain the following theorem.
$$ \begin{aligned}[b] \sum_{n=k}^{\infty }B_{k,n}(x_{1},x_{2}| \lambda )\frac{t^{n}}{n!} &= \sum_{n=k}^{\infty } \binom{n}{k}(x_{1})_{k,\lambda }(1-x_{2})_{n-k, \lambda } \frac{t^{n}}{n!} \\ &=\sum_{n=k}^{\infty }\frac{(x_{1})_{k,\lambda }(1-x_{2})_{n-k, \lambda }}{k!(n-k)!}t^{n} \\ &=\frac{(x_{1})_{k,\lambda }}{k!}t^{k}\sum_{n=0}^{\infty } \frac{(1-x _{2})_{n,\lambda }}{n!}t^{n} \\ &=\frac{(x_{1})_{k,\lambda }}{k!}t^{k}(1+\lambda t)^{\frac{1-x_{2}}{ \lambda }} \\ &=\frac{(x_{1})_{k,\lambda }}{k!}t^{k}e_{\lambda }^{1-x_{2}}(t). \end{aligned} $$
(29)
$$ \begin{aligned} \frac{(x_{1})_{k,\lambda }}{k!}t^{k}(1+ \lambda t)^{\frac{1-x_{2}}{ \lambda }}=\sum_{n=k}^{\infty }B_{k,n}(x_{1},x_{2} |\lambda )\frac{t ^{n}}{n!}, \end{aligned} $$
(30)
$$ \begin{aligned}[b] B_{k,n}(x_{1},x_{2}| \lambda ) &=\binom{n}{k}\bigl(1-(1-x_{1})\bigr)_{n-(n-k), \lambda }(1-x_{2})_{n-k,\lambda } \\ &=B_{n-k,n}(1-x_{2},1-x_{1} |\lambda ). \end{aligned} $$
(31)
$$ \begin{aligned}[b] &\bigl(1-x_{2}-(n-k-1)\lambda \bigr)B_{k,n-1}(x_{1},x_{2}|\lambda )+ \bigl(x_{1}-(k-1) \lambda \bigr)B_{k-1,n-1 }(x_{1},x_{2} |\lambda ) \\ &\quad =\bigl(1-x_{2}-(n-k-1)\lambda \bigr)\binom{n-1}{k}(x_{1})_{k,\lambda }(1-x _{2})_{n-1-k,\lambda } \\ &\qquad {}+\bigl(x_{1}-(k-1)\lambda \bigr)\binom{n-1}{k-1}(x_{1})_{k-1,\lambda }(1-x_{2})_{n-k, \lambda } \\ &\quad =\binom{n}{k}(x_{1})_{k,\lambda }(1-x_{2})_{n-k,\lambda } =B_{k,n}(x _{1},x_{2}|\lambda ) \quad (n,k\in \mathbb{N}). \end{aligned} $$
(32)
Theorem 2.7
For
\(n,k\in \mathbb{N}\), we have
$$ \begin{aligned} &\bigl(1-x_{2}-(n-k-1)\lambda \bigr)B_{k,n-1}(x_{1},x_{2}|\lambda )+ \bigl(x_{1}-(k-1) \lambda \bigr)B_{k-1,n-1 }(x_{1},x_{2} |\lambda ) \\ &\quad =B_{k,n}(x_{1},x_{2}|\lambda ). \end{aligned} $$
If \(f=1\), then we have, from (27),
If \(f(t)=t\), then we also get from (27) that, for \(n\in \mathbb{N}\) and \(x_{1},x_{2}\in \mathbb{Z}_{p}\),
Hence,
By the same method, we get
Note that
Now, we observe that
Thus, by (37), we get
By the same method, we get
Continuing this process, we have
$$ \begin{aligned}[b] \mathbb{B}_{n,\lambda }(1|x_{1},x_{2}) &=\sum_{k=0}^{n}B_{k,n}(x_{1},x _{2}|\lambda ) =\sum_{k=0}^{n} \binom{n}{k}(x_{1})_{k,\lambda }(1-x _{2})_{n-k,\lambda } \\ &=(1+x_{1}-x_{2})_{n,\lambda }. \end{aligned} $$
(33)
$$ \begin{aligned}[b] \mathbb{B}_{n,\lambda }(t|x_{1},x_{2}) &=\sum_{k=0}^{n}\frac{k}{n} \binom{n}{k}(x_{1})_{k,\lambda }(1-x_{2})_{n-k,\lambda } \\ &=(x_{1})_{1,\lambda }(x_{1}+1-\lambda -x_{2})_{n-1,\lambda }. \end{aligned} $$
(34)
$$ \begin{aligned} (x_{1})_{1,\lambda }= \frac{1}{(x_{1}+1-\lambda -x_{2})_{n-1,\lambda }}\mathbb{B}_{n,\lambda }(t|x _{1},x_{2}). \end{aligned} $$
(35)
$$ \begin{aligned}[b] &\mathbb{B}_{n,\lambda } \bigl(t^{2}|x_{1},x_{2}\bigr) \\ &\quad =\frac{1}{n}(x_{1})_{1,\lambda }(1+x_{1}-\lambda -x_{2})_{n-1, \lambda }+\frac{n-1}{n}(x_{1})_{2,\lambda }(1+x_{1}-2 \lambda -x_{2})_{n-2, \lambda }. \end{aligned} $$
(36)
$$ \begin{aligned} \lim_{n\rightarrow \infty } \Bigl(\lim _{\lambda \rightarrow 0}\mathbb{B} _{n,\lambda }\bigl(t^{2}|x,x\bigr) \Bigr) = \lim_{n\rightarrow \infty } \biggl(\frac{x}{n}+ \frac{n-1}{n}x^{2} \biggr)=x^{2}. \end{aligned} $$
$$ \begin{aligned}[b] \sum_{k=1}^{n} \frac{\binom{k}{1}}{\binom{n}{1}}B_{k,n}(x_{1},x_{2}| \lambda ) &= \sum_{k=1}^{n}\binom{n-1}{k-1}(x_{1})_{k,\lambda }(1-x_{2})_{n-k, \lambda } \\ &=\sum_{k=0}^{n-1}\binom{n-1}{k}(x_{1})_{k+1,\lambda }(1-x_{2})_{n-1-k, \lambda } \\ &=(x_{1})_{1,\lambda }(x_{1}+1-\lambda -x_{2})_{n-1,\lambda }. \end{aligned} $$
(37)
$$ \begin{aligned} (x_{1})_{1,\lambda }=\frac{1}{(1+x_{1}-x_{2}-\lambda )}_{n-1,\lambda } \sum_{k=1}^{n}\frac{\binom{k}{1}}{\binom{n}{1}}B_{k,n}(x_{1},x_{2}| \lambda ). \end{aligned} $$
$$ \begin{aligned} (x_{1})_{2,\lambda }=\frac{1}{(1+x_{1}-x_{2}-2\lambda )}_{n-2,\lambda } \sum_{k=2}^{n}\frac{\binom{k}{2}}{\binom{n}{2}}B_{k,n}(x_{1},x_{2}| \lambda ). \end{aligned} $$
$$ \begin{aligned} (x_{1})_{i,\lambda }= \frac{1}{(1+x_{1}-x_{2}-i\lambda )}_{n-i,\lambda }\sum_{k=i}^{n} \frac{\binom{k}{i}}{\binom{n}{i}}B_{k,n}(x_{1},x_{2}| \lambda ) \quad (i\in \mathbb{N}). \end{aligned} $$
(38)
Theorem 2.8
For
\(i\in \mathbb{N}\), we have
$$ \begin{aligned} (x)_{i,\lambda }=\frac{1}{(1+x_{1}-x_{2}-i\lambda )}_{n-i,\lambda } \sum_{k=i}^{n}\frac{\binom{k}{i}}{\binom{n}{i}}B_{k,n}(x_{1},x_{2}| \lambda ) \quad (i\in \mathbb{N}). \end{aligned} $$
Taking the double fermionic p-adic integral on \(\mathbb{Z}_{p}\), we get the following equation:
Therefore, by (39) and Theorem 2.2, we obtain the following theorem.
$$ \begin{aligned} & \int _{\mathbb{Z}_{p} } \int _{\mathbb{Z}_{p} } B_{k,n}(x_{1},x_{2}| \lambda )\,d\mu _{-1}(x_{1})\,d\mu _{-1}(x_{2}) \\ &\quad =\binom{n}{k} \int _{\mathbb{Z}_{p} }(x_{1})_{k,\lambda }\,d\mu _{-1}(x_{1}) \int _{\mathbb{Z}_{p} }(1-x_{2})_{n-k,\lambda }\,d\mu _{-1}(x_{2}). \end{aligned} $$
(39)
Theorem 2.9
For
\(n,k\geq 0\), we have
$$ \begin{aligned} & \int _{\mathbb{Z}_{p} } \int _{\mathbb{Z}_{p} } B_{k,n}(x_{1},x_{2}| \lambda )\,d\mu _{-1}(x_{1})\,d\mu _{-1}(x_{2}) \\ &\quad =\textstyle\begin{cases} \binom{n}{k}\mathcal{E}_{k,\lambda } (2(1)_{n-k,\lambda }+ \mathcal{E}_{n-k,\lambda } ), & \textit{if }n> k, \\ \mathcal{E}_{n,\lambda }, & \textit{if }n=k. \end{cases}\displaystyle \end{aligned} $$
We see from the symmetric properties of two variable degenerate Bernstein polynomials that, for \(n,k\in \mathbb{N}\) with \(n>k\),
Therefore, by Theorem 2.9 and (40), we obtain the following theorem.
$$ \begin{aligned}[b] & \int _{\mathbb{Z}_{p} } \int _{\mathbb{Z}_{p} }B_{k,n}(x_{1},x_{2}| \lambda )\,d\mu _{-1}(x_{1})\,d\mu _{-1}(x_{2}) \\ &\quad =\sum_{l=0}^{k}\binom{n}{k} \binom{k}{l}(-1)^{k+l}(1)_{l,\lambda } \\ & \qquad {}\times \int _{\mathbb{Z}_{p}} \int _{\mathbb{Z}_{p} }(1-x_{1})_{k-l,-\lambda }(1-x_{2})_{n-k,\lambda }\,d\mu _{-1}(x_{1})\,d\mu _{-1}(x_{2}) \\ &\quad =\binom{n}{k} \int _{\mathbb{Z}_{p} }(1-x_{2})_{n-k,\lambda }\,d\mu _{-1}(x _{2}) \Biggl\{ (1)_{k,\lambda }+\sum _{l=0}^{k-1}\binom{k}{l}(-1)^{k+l}(1)_{l, \lambda } \mathcal{E}_{k-l,-\lambda } (2) \Biggr\} \\ &\quad =\binom{n}{k}\mathcal{E}_{n-k,\lambda }(2) \Biggl\{ (1)_{k,\lambda }+ \sum_{l=0}^{k-1}\binom{k}{l}(-1)^{k+l}(1)_{l,\lambda } \mathcal{E}_{k-l,- \lambda }(2) \Biggr\} . \end{aligned} $$
(40)
Theorem 2.10
For
\(k\in \mathbb{N}\), we have
$$ \begin{aligned} \mathcal{E}_{k,\lambda }=(1)_{k,\lambda }+ \sum_{l=0}^{k-1} \binom{k}{l}(-1)^{k+l}(1)_{l,\lambda } \bigl(\mathcal{E}_{k-l,-\lambda }+2(1)_{k-l,-\lambda } \bigr). \end{aligned} $$
Note that
$$ \begin{aligned}[b] &\sum_{l=0}^{k-1} \binom{k}{l}(-1)^{k+l}(1)_{l,\lambda }(1)_{k-l,- \lambda } \\ &\quad =(-1)^{k} \Biggl(\sum_{l=0}^{k} \binom{k}{l}(-1)_{l,-\lambda }(1)_{k-l,- \lambda }-(-1)_{k,-\lambda } \Biggr) \\ &\quad =(-1)^{k} \bigl((0)_{k,-\lambda }-(-1)_{k,-\lambda } \bigr) \\ &\quad =-(1)_{k,\lambda }. \end{aligned} $$
Corollary 2.11
For
\(k\in \mathbb{N}\), we have
$$ \begin{aligned} \mathcal{E}_{k,\lambda }=-(1)_{k,\lambda }+ \sum_{l=0}^{k-1} \binom{k}{l}(-1)^{k+l}(1)_{l,\lambda } \mathcal{E}_{k-l,-\lambda }. \end{aligned} $$
3 Conclusions
In [1, 2], Carlitz initiated the study of degenerate versions of some special polynomials and numbers, namely the degenerate Bernoulli and Euler polynomials and numbers. Here we would like to draw the attention of the reader to the fact that Kim et al. have introduced various degenerate polynomials and numbers and investigating their properties, some identities related to them and their applications by means of generating functions, combinatorial methods, umbral calculus, p-adic analysis and differential equations (see [8, 9] and the references therein). It is amusing that this line of study led them even to the introduction of degenerate gamma functions and degenerate Laplace transforms (see [7]). These already demonstrate that studying various degenerate versions of known special numbers and polynomials can be very promising and rewarding. Furthermore, we can hope that many applications will be found not only in mathematics but also in sciences and engineering.
In this paper, we investigated some properties and identities for degenerate Euler polynomials in connection with degenerate Bernstein polynomials and operators which were recently introduced as degenerate versions of the classical Bernstein polynomials and operators. This has been done by means of fermionic p-adic integrals on \(\mathbb{Z}_{p}\) and generating functions. In addition, we studied two variable degenerate Bernstein polynomials and the degenerate Bernstein operators.
Acknowledgements
The fourth author’s work in this paper was conducted during the sabbatical year of Kwangwoon University in 2018.
Competing interests
The authors declare that they have no competing interests.
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