Some identities of Bernoulli numbers and polynomials associated with Bernstein polynomials

We investigate some interesting properties of Bernstein polynomials associated with boson p-adic integrals on Zp.


Introduction
are called the Bernstein basis polynomials (or the Bernstein polynomials of degree n) (see [10]). Recently, Acikgoz and Araci have studied the generating function for Bernstein polynomials (see [1,2]). Their generating function for B k,n (x) is given by where k = 0, 1, . . . and x ∈ [0, 1]. Note that if n < k for n = 0, 1, . . . (see [1,2]). The Bernstein polynomials can also be defined in many different ways. Thus, recently, many applications of these polynomials have been looked for by many authors. Some researchers have studied the Bernstein polynomials in the area of approximation theory (see [1,2,3,7,9,10]). In recent years, Acikgoz and Araci [1,2] have introduced several type Bernstein polynomials.
In the present paper, we introduce the Bernstein polynomials on the ring of padic integers Z p . We also investigate some interesting properties of the Bernstein polynomials related to the bosonic p-adic integrals on the ring of p-adic integers Z p .

Bernstein polynomials related to the bosonic p-adic integrals on Z p
Let p be a fixed 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 numbers and the completion of the algebraic closure of Q p , respectively. Let v p be the normalized exponential [4]). We shall write dµ 1 (x) to remind ourselves that x is the variable of integration. Let U D(Z p ) be the space of uniformly differentiable function on Z p . Then µ 1 yields the fermionic p-adic q-integral of a function f ∈ U D(Z p ) : [4]). Many interesting properties of (2.2) were studied by many authors (cf. [4,8] and the references given there). For n ∈ N, write f n (x) = f (x + n). We have This identity is to derives interesting relationships involving Bernoulli numbers and polynomials. Indeed, we note that where B n (x) are the Bernoulli polynomials (cf. [4]). From (1.2), we have By (2.5) and (2.6), we obtain the following proposition.
From (2.4), we note that (2.7) B n (2) = (B(1) + 1) n − n = (B + 1) n = B n , n > 1 with the usual convention of replacing B n by B n . Thus, we have . Therefore we obtain the following theorem.
And also we obtain Therefore we obtain the following result.
From the property of the Bernstein polynomials of degree n, we easily see that Continuing this process, we obtain the following theorem.