On the q-Euler numbers related to modified q-Bernstein polynomials

In this paper we investigate some interesting formulae of q-Euler numbers and polynomials related to the modified q-Bernstein polynomials.


Introduction
are called the Bernstein basis polynomials (or the Bernstein polynomials of degree n) (see [17]). 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 (cf. [5,6,7,12]). Let N be the natural numbers and Z + = N ∪ {0}. Let U D(Z p ) be the space of uniformly differentiable function on Z p .
Let q ∈ C p with |1 − q| p < p −1/(p−1) and x ∈ Z p . Then q-Bernstein type operator for f ∈ U D(Z p ) is defined by (see [14,15]) for k, n ∈ Z + , where is called the modified q-Bernstein polynomials of degree n. When we put q → 1 in n−k and we obtain the classical Bernstein polynomial, defined by (1.2). We can deduce very easily from (1.5) that (see [14]). For 0 ≤ k ≤ n, derivatives of the nth degree modified q-Bernstein polynomials are polynomials of degree n − 1 : (see [14]). 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 recent years, the q-Bernstein polynomials have been investigated and studied by many authors in many different ways (see [14,15,17] and references therein [4,16]). In [16], 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 [1,2], Acikgoz and Araci have introduced several type Bernstein polynomials. The Acikgoz and Araci paper to announce in the conference is actually motivated to write this paper. In [17], Simsek and Acikgoz constructed a new generating function of the q-Bernstein type polynomials and established elementary properties of this function. In [14], Kim, Jang and Yi proposed the modified q-Bernstein polynomials of degree n, which are different q-Bernstein polynomials of Phillips. In [15], Kim, Choi and Kim 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 Z p .

q-Euler numbers and polynomials related to the fermionic p-adic integrals on Z p
For N ≥ 1, the fermionic q-extension µ q of the p-adic Haar distribution µ Haar : [5,8]). We shall write dµ −q (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 −q yields the fermionic p-adic q-integral of a function f ∈ U D(Z p ) : [8,9,10,13]). Many interesting properties of (2.2) were studied by many authors (see [8,9] and the references given there). Using (2.2), we have the fermionic p-adic invariant integral on Z p as follows: For n ∈ N, write f n (x) = f (x + n). We have This identity is obtained by Kim in [8] to derives interesting properties and relationships involving q-Euler numbers and polynomials. For n ∈ Z + , we note that where E n,q are the q-Euler numbers (see [11]). It is easy to see that E 0,q = 1. For n ∈ N, we have From this formula, we have the following recurrence formula with the usual convention of replacing E l by E l,q . By the simple calculation of the fermionic p-adic invariant integral on Z p , we see that where n l = n!/l!(n − l)! = n(n − 1) · · · (n − l + 1)/l!. Now, by introducing the following equations: This identity is a peculiarity of the p-adic q-Euler numbers, and the classical Euler numbers do not seem to have a similar relation. Let F q (t) be the generating function of the q-Euler numbers. Then we obtain From (2.11) we note that It is well-known that where E n,q (x) are the q-Euler polynomials (see [11]). In the special case x = 0, the numbers E n,q (0) = E n,q are referred to as the q-Euler numbers. Thus we have It is easily verified, using (2.12) and (2.13), that the q-Euler polynomials E n,q (x) satisfy the following formula: Using formula (2.15) when q tends to 1, we can readily derive the Euler polynomials, (see [8]). Note that E n (0) = E n are referred to as the nth Euler numbers. Comparing the coefficients of t n /n! on both sides of (2.15), we have We refer to [n] q as a q-integer and note that [n] q is a continuous function of q.
In an obvious way we also define a q-factorial, and a q-analogue of binomial coefficient (cf. [10,11]). Note that It readily follows from (2.17) that [11,12]). It can be readily seen that Thus by (2.13) and (2.19), we have From now on, we use the following notation , n ∈ Z + (see [12] Therefore, we obtain the following theorem. By (2.19) and simple calculation, we find that Therefore, we deduce the following theorem.  By (2.16) and Corollary 2.3, we obtain the following corollary.
It is easy to see that (cf. [12]). From (2.25) and Corollary 2.4, we can also derive the following interesting formula for q-Euler polynomials.
These polynomials are related to the many branches of Mathematics, for example, combinatorics, number theory, discrete probability distributions for finding higher-order moments (cf. [10,11,13]). By substituting x = 0 into the above, we have where E n,q is the q-Euler numbers.
3. q-Euler numbers, q-Stirling numbers and q-Bernstein polynomials related to the fermionic p-adic integrals on Z p First, we consider the q-extension of the generating function of Bernstein polynomials in (1.3).
For q ∈ C p with |1 − q| p < p −1/(p−1) , we obtain which is the generating function of the modified q-Bernstein type polynomials (see [15]). Indeed, this generating function is also treated by Simsek and Acikgoz (see [17]). Note that lim q→1 F (k) t, x). It is easy to show that From (1.4), (2.3), (2.15) and (3.2), we derive the following theorem.
where E n,q are the q-Euler numbers.
It is possible to write [x] k q as a linear combination of the modified q-Bernstein polynomials by using the degree evaluation formulae and mathematical induction. Therefore we obtain the following theorem.
Theorem 3.3. For k, n ∈ Z + and i ∈ N with i − 1 ≤ n, The q-String numbers of the first kind is defined by (1 + [k] q z) = n k=0 S 1 (n, k; q)z k , and the q-String number of the second kind is also defined by S 2 (n, k; q)z k (see [15]). Therefore, we deduce the following theorem.
By Theorem 3.2, Theorem 3.4 and the definition of fermionic p-adic integrals on Z p , we obtain the following theorem.
Theorem 3.5. For k, n ∈ Z + and i ∈ N, where E i,q is the q-Euler numbers.
Let i − 1 ≤ n. It is easy to show that From (3.6) and Theorem 3.2, we have the following theorem.
Theorem 3.6. For k, n ∈ Z + and i ∈ N, where E i,q are the q-Euler numbers.
In the same manner, we can obtain the following theorem.
Theorem 3.7. For k, n ∈ Z + and i ∈ N, where E i,q are the q-Euler numbers.

Further remarks and observations
The q-binomial formulas are known, For h ∈ Z, n ∈ Z + and r ∈ N, we introduce the extended higher-order q-Euler polynomials as follows [11]: Then (4.3) Let us now define the extended higher-order Nörlund type q-Euler polynomials as follows [11]: .
In the special case are called the extended higher-order Nörlund type q-Euler numbers. From (4.4), we note that (4.5) A simple manipulation shows that Formulas (4.5), (4.6) and (4.7) imply the following lemma.