On q-Euler Numbers Related to the Modified q-Bernstein Polynomials

and Applied Analysis 3 see 8 . For 0 ≤ k ≤ n, derivatives of the nth degree modified q-Bernstein polynomials are polynomials of degree n − 1: d dx Bk,n ( x, q ) n ( qBk−1,n−1 ( x, q ) − q1−xBk,n−1 ( x, q )) ln q q − 1 1.9 see 8 . 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. In the recent years, the q-Bernstein polynomials have been investigated and studied by many authors in many different ways see 1, 8, 9 and references therein 10, 11 . In 11 , Phillips gave many results concerning the q-integers and an account of the properties of q-Bernstein polynomials. He gave many applications of these polynomials on approximation theory. In 2, 3 , Acikgoz and Araci have introduced several type Bernstein polynomials. The Acikgoz and Araci paper to announced in the conference is actually motivated to write this paper. In 1 , Simsek and Acikgoz constructed a new generating function of the q-Bernstein type polynomials and established elementary properties of this function. In 8 , Kim et al. proposed the modified q-Bernstein polynomials of degree n,which are different q-Bernstein polynomials of Phillips. In 9 , Kim et al. investigated some interesting properties of the modified q-Bernstein polynomials of degree n related to q-Stirling numbers and Carlitz’s q-Bernoulli numbers. In the present paper, we consider q-Euler numbers, polynomials, and q-Stirling numbers of first and second kinds. We also investigate some interesting properties of the modified q-Bernstein polynomials of degree n related to q-Euler numbers and q-Stirling numbers by using fermionic p-adic integrals on Zp. 2. q-Euler Numbers and Polynomials Related to the Fermionic p-Adic Integrals on Zp For N ≥ 1, the fermionic q-extension μq of the p-adic Haar distribution μHaar, μ−q ( a pZp ) −qa [ pN ] −q , 2.1 is known as a measure on Zp, where a pZp {x ∈ Qp | |x − a|p ≤ p−N} cf. 4, 12 . We will write dμ−q x to remind ourselves that x is the variable of integration. Let UD Zp be the space of uniformly differentiable function on Zp. Then μ−q yields the fermionic p-adic q-integral of a function f ∈ UD Zp : I−q ( f ) ∫ Zp f x dμ−q x lim N→∞ 1 q 1 qpN pN−1 ∑ x 0 f x −qx 2.2 4 Abstract and Applied Analysis cf. 12–15 . Many interesting properties of 2.2 were studied by many authors see 12, 13 and the references given there . Using 2.2 , we have the fermionic p-adic invariant integral on Zp as follows: lim q→−1 Iq ( f ) I−1 ( f ) ∫ Zp f a dμ−1 x . 2.3 For n ∈ N, write fn x f x n . We have I−1 ( fn ) −1 I−1 ( f ) 2 n−1 ∑ l 0 −1 n−l−1f l . 2.4 This identity is obtained by Kim in 12 to derive interesting properties and relationships involving q-Euler numbers and polynomials. For n ∈ Z ,we note that