A NOTE ON DEGENERATE BERNSTEIN AND DEGENERATE EULER POLYNOMIALS

In this paper, we investigate the recently introduced degenerate Bernstein polynomials and operators and derive some of their properties. Also, we give some properties of the degenerate Euler numbers and polynomials and their connection with the degenerate Euler polynomials.

Here B n , (n ≥ 0), is called Bernstein operator of order n.
A Bernoulli trial involves performing an experiment once and noting whether a particular event A occurs.The outcome of Bernoulli trial is said to be "Success" if A occurs and a "failure" otherwise.The probability P n (k) of k successes in n independent Bernoulli trials with the probability of success p is given by the binomial probability law From the definition of Bernstein polynomials, we note that Bernstein basis is the probability mass function of binomial distribution.The Bernstein polynomials of degree n can be defined by blending together two Bernstein polynomials of degree n − 1.That is, the k-th Bernstein polynomial of degree n can be written as Thus, we note that (1.4) For λ ∈ R, L. Carlitz introduced the degenerate Euler poynomials given by the generating function 2 t n n! , (see [4]).

Degenerate Bernstein polynomials and operators
Let f (x) be a continuous function on [0, 1].Then the degenerate Bernstein operator of order n is defined as where x ∈ [0, 1] and n, k ∈ Z ≥0 .From (2.1), we note that Now, we observe that By comparing the coefficients on the both sides of (2.3), we get From (2.2) and (2.4), we have Also, we get from (2.1) that for f (x) = x, (2.6) From (2.6), we can derive the following equation (2.7).
where (x Therefore, we obtain the following theorem.
Theorem 2.1.For n ≥ 0, we have Let f, g be continuous functions defined on [0, 1].Then we note that where α, β are constants.So, the degenerate Bernstein operator is linear.From (1.8), we note that It is easy to show that Thus we get From (2.1), we have (2.10) It is easy to show that From (2.10) and (2.12), we have (2.13) Therefore, by (2.13), we obtain the following theorem.
From (1.8) and (2.5), it is to see that In the same manner, we can show that n−i,λ .

Degenerate Euler polynomials associated with degenerate Bernstein polynomials
From (1.5), we note that By comparing the coefficients on both sides of (3.1), we obtain the following theorem.
Theorem 3.1.For n ≥ 0, we have From Theorem 4, we note that By (1.5), we get (1 + λt) By comparing the coefficients on the sides of (3.2), we have In particular, if we take x = −1, we get Therefore, we obtain the following theorem.
0) are called the degenerate Euler numbers.It is easy to show that lim λ→0 E n,λ (x) = E n (x), where E n (x) are the Euler polyno-