Some identities of degenerate Euler polynomials associated with degenerate Bernstein polynomials

In this paper, we investigate some properties and identities for degenerate Euler polynomials in connection with degenerate Bernstein polynomials by means of fermionic p-adic integrals on Zp and generating functions. In addition, we study two variable degenerate Bernstein polynomials and the degenerate Bernstein operators.


Introduction
Let p be a fixed odd prime number. Throughout this paper, Z p , Q p and C p , will 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 ν p be the normalized exponential valuation of C p with |p | p = p −νp(p) = 1 p . For λ ∈ C p with |λ| p < p − 1 p−1 , the degenerate Euler polynomials are defined by the generating function 2 (1 + λt) 1 λ + 1 (1 + λt) x λ = ∞ n=0 E n,λ (x) t n n! , (see [1,2]).
Thus, by (5), we get where n, k are nonnegative integers. Let f be a continuous function on Z p . Then the degenerate Bernstein operator of order n is given by , (see [4,5]).
The fermionic p-adic integral on Z p is defined by Kim as [3,5]).
From (8), we note that On the other hand, By (10) and (11), we get Zp (x + y) n,λ dµ −1 (y) = E n,λ (x), (n ≥ 0), (see [4,5]). (12) In this paper, we investigate some properties and identities for degenerate Euler polynomials in connection with degenerate Bernstein polynomials by means of fermionic p-adic integrals on Z p and generating functions. In addition, we study two variable degenerate Bernstein polynomials and the degenerate Bernstein operators.

Degenerate Euler and Bernstein polynomials
From (1), we note that Comparing the coefficients on both sides of (13), we have where δ n,k is the Kronecker's symbol. By (1), we easily get From (1), (4) and (14), we note that where n is a positive integer. Therefore, by (16), we obtain the following theorem.
For k ∈ N, the higher order degenerate Euler polynomials are given by the generating function 2 (1 + λt) From (5) and (25), we note that Therefore, by comparing the cofficients on both sides of (26), we obtain the following theorem.
Theorem 2.6. For n, k ∈ N, we have Let f be a continuous function on Z p . For x 1 , x 2 ∈ Z p , we consider the degenerate Bernstein operator of order n given by where where n, k are nonnegative integers.
Here, B k,n (x 1 , x 2 |λ) 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.
If f = 1, then we have from (27), If f (t) = t, then we also get from (27) that for n ∈ N and x 1 , 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 (38) Taking double fermionic p-adic integral on Z p , we get the following equation: Therefore, by (39) and Theorem 2.2, we obtain the following theorem.
We get from the symmetric properties of two variable degenerate Bernstein polynomials that for n, k ∈ N with n > k, Therefore, by Theorem 2.9 and (40), we obtain the following theorem.
Theorem 2.10. For k ∈ N, we have Note that Corollary 2.11. For k ∈ N, we have

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 readers that T. Kim and his colleagues have been introducing 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 [4,5] 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 Z p and generating functions. In addition, we studied two variable degenerate Bernstein polynomials and the degenerate Bernstein operators.