Some Identities on Degenerate Bernstein and Degenerate Euler Polynomials

In recent years, intensive studies on degenerate versions of various special numbers and polynomials have been done by means of generating functions, combinatorial methods, umbral calculus, p-adic analysis and differential equations. The degenerate Bernstein polynomials and operators were recently introduced as degenerate versions of the classical Bernstein polynomials and operators. Herein, we firstly derive some of their basic properties. Secondly, we explore some properties of the degenerate Euler numbers and polynomials and also their relations with the degenerate Bernstein polynomials.


Introduction
Let us denote the space of continuous functions on [0, 1] by C[0, 1], and the space of polynomials of degree ≤ n by P n .The Bernstein operator B n of order n, (n ≥ 1), associates to each f ∈ C[0, 1] the polynomial B n ( f |x) ∈ P n , and was introduced by Bernstein as (see [1,2]): (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14]) where are called either Bernstein polynomials of degree n or Bernstein basis polynomials of degree n.The Bernstein polynomials of degree n can be defined in terms of two such polynomials of degree n − 1.That is, the k-th Bernstein polynomial of degree n can be written as From (2), the first few Bernstein polynomials B k,n (x) are given by Thus, we note that .
Recently, the degenerate Bernstein polynomials of degree n are introduced as (see [8]) From (7), it is not difficult to show that the generating function for B k,n (x|λ) is given by (see [8]) By (8), we easily get lim λ→0 B k,n (x|λ) = B k,n (x), (n, k ≥ 0).The Bernstein polynomials are the mathematical basis for Bézier curves which are frequently used in computer graphics and related fields.In this paper, we investigate the degenerate Bernstein polynomials and operators.We study their elementary properties (see also [8]) and then their further properties in association with the degenerate Euler numbers and polynomials.
Finally, we would like to briefly go over some of the recent works related with Bernstein polynomials and operators.
Kim-Kim in Ref. [19] gave identities for degenerate Bernoulli polynomials and Korobov polynomials of the first kind.The authors in Ref. [20] introduced a generalization of the Bernstein polynomials associated with Frobenius-Euler polynomials.The paper [21] deals with some identities of q-Euler numbers and polynomials associated with q-Bernstein polynomials.In Ref. [22], the authors studied a space-time fractional diffusion equation with initial boundary conditions and presented a numerical solution for that.Both normalized Bernstein polynomials with collocation and Galerkin methods are applied to turn the problem into an algebraic system.Kim in Ref. [23] introduced some identities on the q-integral representation of the product of the several q-Bernstein type polynomials.Grouped data are commonly encountered in applications.In Ref. [24], Kim-Kim studied some properties on degenerate Eulerian numbers and polynomials.The authors in Ref. [25] give an overview of several results related to partially degenerate poly-Bernoulli polynomials associated with Hermit polynomials.

Degenerate Bernstein Polynomials and Operators
The degenerate Bernstein operator of order n is defined, for f ∈ C[0, 1], as where x ∈ [0, 1] and n, k ∈ Z ≥0 .
Theorem 1.For n ≥ 0, we have and Proof.From ( 9), we clearly have Now, we observe that Comparing the coefficients on both sides of (11), we derive Combining (10) with (12), we have Furthermore, we get from (9) that for f (x) = x, From ( 14), we can easily deduce the following Equation ( 15): Let f , g be continuous functions defined on [0, 1].Then, we clearly have where α, β are constants.So, the degenerate Bernstein operator is linear.From (7), we note that This shows that we have Theorem 2. For f ∈ C[0, 1] and n ∈ Z ≥0 , we have Proof.From ( 9), it is immediate to see that We need to note the following: and Let k + j = m.Then, by ( 19), we obviously have Combining ( 18) with ( 19)- (21) gives the following result: Theorem 3.For n, k ∈ Z ≥0 and x ∈ [0, 1], we have Proof.From ( 7), (17), and ( 20), we observe that

Degenerate Euler Polynomials Associated with Degenerate Bernstein Polynomials
Theorem 4. For n ≥ 0, the following holds true: Proof.From (4), we remark that The result follows by comparing the coefficients on both sides of (23).