Some Identities of Fully Degenerate Bernoulli Polynomials Associated with Degenerate Bernstein Polynomials

: In this paper, we investigate some properties and identities for fully degenerate Bernoulli polynomials in connection with degenerate Bernstein polynomials by means of bosonic p -adic integrals on Z p and generating functions. Furthermore, we study two variable degenerate Bernstein polynomials and the degenerate Bernstein operators.

Recall that the Daehee polynomials are defined by the generating function to be and for x = 0, D n = D n (0) are called the Daehee numbers (see [10,11]).
Also, the higher order Daehee polynomials are defined by the generating function to be and for n (0) are called the higher order Daehee numbers. From (10), we observe By (10) and (14), we get From (3) and (10), we observe that By (17), we get From (1) and (3), we note that Comparing the cofficients on both sides of (19), we get where δ k,n is the Kronecker's symbol.
By (4) and (20), we have The generating function of fully degenerate Bernoulli polynomials introduced in (5) can be expressed as bosonic p-adic integral but the generating function of degenerate Bernoulli polynomials introduced in (1) is not expressed as a bosonic p-adic integral. This is why we considered the fully degenerate Bernoulli polynomials, and the motivation of this paper is to investigate some identities of them associated with degenerate Bernstein polynomials.
In this paper, we consider the fully degenerate Bernoulli polynomials and investigate some properties and identities for these polynomials in connection with degenerate Bernstein polynomials by means of bosonic p-adic integrals on Z p and generating functions. Furthermore, we study two variable degenerate Bernstein polynomials and the degenerate Bernstein operators.

Fully Degenerate Bernoulli and Bernstein Polynomials
From (10), we observe that From (22), we obtain the following Lemma.

Theorem 2.
For n ∈ N, we have Corollary 1. For n ∈ N, we have By (17), we get In [8], we note that By (33) and (34) we get Therefore, by (35), we obtain the following theorem.
For k ∈ N, the higher-order fully degenerate Bernoulli polynomials are given by the generating function (See [8,12,13]). When x = 0, B (k) n (x|0) are called the higher-order fully degenerate Bernoulli numbers. From (5) and (38), we note that and hence, we get Therefore, by (39) and (40), 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 5, we obtain the following theorem.

Remark
Let us assume that the probability of success in an experiment is p. We wondered if we could say the probability of success in the 9th trial is still p after failing eight times in a ten trial experiment, because there is a psychological burden to be successful. It seems plausible that the probability is less than p. The degenerate Bernstein polynomial B n (x|λ) is used in the probability of success. Thus, we give examples in our results as follows: Example 1. Let n = 2, we have Example 2. Let n = 1, we have Example 3. Let n = 1, k = 2, we have

Conclusions
In this paper, we studied the fully degenerate Bernoulli polynomials associated with degenerate Bernstein polynomials. In Section 1, Equations (12), (18), (20) and (21) are some properties of them. In Section 2, Theorems 1-3 are results of identities for fully degenerate Bernoulli polynomials in connection with degenerate Bernstein polynomials by means of bosonic p-adic integrals on Z p and generating functions. Theorems 4-6 are results of higher-order fully Bernoulli polynomials in connection with two variable degenerate Bernstein polynomials by means of bosonic p-adic integrals on Z p and generating functions.