Some Properties on the q-Euler Numbers and Polynomials

and Applied Analysis 3 is to give some interesting new identities for the q-Euler numbers and polynomials by using the fermionic p-adic integral on Zp and 1.7 . 2. Some Identities on q-Euler Polynomials From 1.4 , we note that ∫ Zp e x y z qdμ−1 z e x y t 2 qet 1 ∞ ∑ n 0 En,q ( x y ) t n! . 2.1 Thus, by 1.4 and 2.1 , we get


Introduction
Let p be a fixed odd prime number.Throughout this paper Z p , Q p , and C p will denote the ring of p-adic rational integers, the field of p-adic rational numbers, and the completion of the algebraic closure of Q p .The p-adic absolute value | • | p is defined by |p| p 1/p.In this paper, we assume that q ∈ C p with |1 − q| p < 1.As is well known, the fermionic p-adic integral on Z p is defined by Kim as follows: where f ∈ C Z p the space of continuous functions on Z p see 1 .From 1.1 , we note that The q-Euler polynomials are defined by with the usual convention about replacing E n q by E n,q see 2, 3 .Let us take f y q y e t x y .Then, by 1.2 , we get By 1.3 and 1.4 , we get the Witt's formula for the q-Euler polynomials as follows: In the special case, x 0, E n,q 0 E n,q are called the n-th q-Euler numbers.From 1.3 , we can derive the following recurrence relation for the q-Euler numbers E n,q : with the usual convention about replacing E n q by E n,q see 4 .By 1.5 , we easily see that where Cohen introduced many interesting and valuable identities related to Euler and Bernoulli numbers and polynomials in his book see 14 .In 13 , Ryoo has introduced the q-Euler numbers and polynomials with weight α, and Simsek et al. have studied q-Euler numbers and polynomials, and they introduced many interesting identities and properties see 3, 15, 16 .In this paper, we consider the q-Euler numbers and polynomials with weight α 1.By applying the fermionic p-adic integral on Z p , we derive many not only new but also some interesting identities on the q-extension of Euler numbers and polynomials.In particular, we consider that Theorems 2.5, 2.6, 2.7, and 2.9 are important identities because these identities are closely related to Frobenius-Euler numbers and polynomials.As is well known, Frobenius-Euler numbers and polynomials are important to study p-adic l-functions in the number theory and mathematical physics related to fermionic distributions.In 17 , Bayad and Kim have studied some interesting identities and properties on the q-Euler numbers and polynomials associated with Bernstein polynomials.Recently, several authors have studied some properties of q-Euler numbers and polynomials see 1-19 .The purpose of this paper Abstract and Applied Analysis 3 is to give some interesting new identities for the q-Euler numbers and polynomials by using the fermionic p-adic integral on Z p and 1.7 .

Some Identities on q-Euler Polynomials
From 1.4 , we note that Thus, by 1.4 and 2.1 , we get where n ∈ Z .

2.2
By 2.2 , we get

2.3
From 2.3 , we can derive the following equation 2.4 : Therefore, by 2.4 , we obtain the following theorem.

2.5
Let us replace y by −y in Theorem 2.1.Then we get Thus, we have Therefore, by Theorem 2.1 and 2.7 , we obtain the following corollary.
Corollary 2.2.For n ∈ Z , one has From 2.2 , we have Therefore, by 2.3 and 2.9 , we obtain the following theorem.

2.10
Letting y 1 in Theorem 2.1, we see that Abstract and Applied Analysis 5

2.11
Therefore, by 2.11 , we obtain the following theorem.
Theorem 2.4.For n ∈ Z , one has Replacing y by 1 and n by 2n in Corollary 2.2, we have

2.13
Therefore, by 2.13 , we obtain the following theorem.
Theorem 2.5.For n ∈ Z , one has Replacing y by 1 and n by 2n in Theorem 2.3, we have Abstract and Applied Analysis

2.15
Therefore, by 2.15 , we obtain the following theorem.

2.16
Replacing y by 1/2 and n by 2n in Theorem 2.3, we get 8n 4 .

2.17
Thus, by 2.17 , we get Note that

2.19
Therefore, by 2.18 and 2.19 , we obtain the following theorem.
Theorem 2.7.For n ∈ N, one has Replacing y by 1 and n by 2n 1 in Corollary 2.2, we see that

2.23
Therefore, by 2.23 , we obtain the following theorem.