On the q-Euler numbers and polynomials with weight 0

The purpose of this paper is to investigate some properties of q-Euler numbers and polynomials with weight 0. From those q-Euler numbers with weight 0, we derive some identities on the q-Euler numbers and polynomials with weight 0.

By (1), (2) and (3), we see that In the special case, n = 1, we get where H n (−q −1 ) are the n-th Frobenius-Euler numbers. From (5), we note that the q-Euler numbers with weight 0 are given bỹ Therefore, by (6), we obtain the following theorem.
where H n (−q −1 ) are called the n-th Frobenius-Euler numbers.
Let us define the generating function of the q-Euler numbers with weight 0 as follows:F Then, by (4) and (7), we get Now we define the q-Euler polynomials with weight 0 as follows: Thus, (5) and (9), we get where H n (−q −1 , x) are called the n-th Frobenius-Euler polynomials (see [9]). Therefore, by (11), we obtain the following theorem.
From (3) and Theorem 2, we note that where n ∈ N with n ≡ 1 (mod 2). Therefore, by (12), we obtain the following corollary.
Corollary 3. For n ∈ N, with n ≡ 1 (mod 2) and m ∈ Z + , we have In particular, q = 1, we get E m (n) + E m = 2 n−1 l=0 (−1) l l m , where E m and E m (n) are called the m-th Euler numbers and polynomials which are defined by By (3), we easily see that Thus, by (13), we get Therefore, by (13), we obtain the following theorem.
Theorem 4. For n ∈ Z + , we have where H n (−q −1 , x) are called the n-th Frobenius-Euler polynomials and H n (−q −1 ) are called the n-th Frobenius-Euler numbers. In particular, q = 1, we have where E n are called the n-th Euler numbers.
Therefore, by (17), we obtain the following theorem.
Theorem 7. For n ∈ N, we have Let C(Z p ) be the space of continuous functions on Z p . For f ∈ C(Z p ), p-adic analogue of Bernstein operator of order n for f is given by where n, k ∈ Z + (see [1,6,7]). For n, k ∈ Z + , p-adic Bernstein polynomials of degree n is defined by [1,6,7]).
Let us take the fermionic p-adic q-integral on Z p for one Bernstein polynomials in (22) as follows: Therefore, by (23) and (24), we obtain the following theorem.
By Theorem 1 and Theorem 2, we get where H n (−q) are called the n-th Frobenius-Euler numbers.