Explicit Formulas involving q-Euler Numbers and Polynomials

In this paper, we deal with q-Euler numbers and q-Bernoulli numbers. We derive some interesting relations for q-Euler numbers and polynomials by using their generating function and derivative operator. Also, we show between the q-Euler numbers and q-Bernoulli numbers via the p-adic q-integral in the p-adic integer ring.


PRELIMINARIES
Imagine that p be a fixed odd prime number. Throughout this paper we use the following notations, by Z p denotes the ring of p-adic rational integers, Q denotes the field of rational numbers, Q p denotes the field of p-adic rational numbers, and C p denotes the completion of algebraic closure of Q p . Let N be the set of natural numbers and N * = N ∪ {0}.
The p-adic absolute value is defined by In this paper we assume |q − 1| p < 1 as an indeterminate.
We say that f is a uniformly differentiable function at a point a ∈ Z p , if the difference quotient x − y has a limit f´(a) as (x, y) → (a, a) and denote this by f ∈ U D (Z p ).
Let U D (Z p ) be the set of uniformly differentiable function on Z p . For f ∈ U D (Z p ), let us start with the expressions 2000 Mathematics Subject Classification. Primary 05A10, 11B65; Secondary 11B68, 11B73.
Key words and phrases. Euler numbers and polynomials, q-Euler numbers and polynomials with weight 0, q-Bernoulli numbers with weight 0, p-adic q-integral.
represents p-adic q-analogue of Riemann sums for f . The integral of f on Z p will be defined as the limit (N → ∞) of these sums, when it exists. The p-adic q-integral of function f ∈ U D (Z p ) is defined by T. Kim The bosonic integral is considered as a bosonic limit q → 1, I 1 (f ) = lim q→1 I q (f ). Similarly, the fermionic p-adic integral on Z p is introduced by T. Kim as follows: (for more details, see [9][10][11][12]). In [6], the q-Euler polynomials with wegiht 0 are introduced as where E n,q (0) = E n,q are called q-Euler numbers with weight 0. Then, q-Euler numbers are defined as with the usual convention about replacing E q n by E n,q is used. Similarly, the q-Bernoulli polynomials and numbers with weight 0 are defined, respectively and B n,q = Zp y n dµ q (y) (for more informations, see [4]). We, by using Kim's et al. method in [2], will investigate some interesting identities on the q-Euler numbers and polynomials from their generating function and derivative operator. Consequently, we derive some properties on q-Euler numbers and polynomials and q-Bernoulli numbers and polynomials by using q-Volkenborn integral and fermionic p-adic q-integral on Z p .

ON KIM'S q-EULER NUMBERS AND POLYNOMIALS
Let us consider Kim's q-Euler polynomials as follows: Here x is a fixed parameter. Thus, by expression of (2.1), we can readily see the following Last from equality, taking derivative operator D as D = d dt on the both sides of (2.2). Then, we easily see that where k ∈ N * and I is identity operator. By multiplying e −t on both sides of (2.3), we get Let us take derivative operator D m (m ∈ N) on both sides of (2.4). Then we get Let G [0] (not G (0)) be the constant term in a Laurent series of G (t). Then, from (2.5), we get By expressions of (2.6) and (2.7), we see that From (2.1), we note that By (2.9), we easily see, Now, let us consider definition of integral from 0 to 1 in (2.8), then we have where B (m, n) is beta function which is defined by , m > 0 and n > 0.
As a result, we obtain the following theorem Substituting m = k + 1 into Theorem 1, we readily get By (2.1), it follows that max{k,m} Last from equality, we discover the following (2.12) Here [.] is Gauss' symbol. Then, taking integral from 0 to 1 both sides of last equality, we get Consequently, we derive the following theorem Theorem 2. The following identity is true.
In view of (2.1) and (2.12), we discover the following applications: By expressions (2.12) and (2.14), we have the following Theorem Theorem 3. For k ∈ N, we have . p-adic integral on Z p associated with Kim's q-Euler polynomials In this section, we consider Kim's q-Euler polynomials by means of p-adic qintegral on Z p . Now we start with the following assertion.
Let m, k ∈ N, Then by (2.8), On the other hand, right hand side of (2.8), Equating I 1 and I 2 , we get the following theorem Let us take fermionic p-adic q-inetgral on Z p left hand side of (2.15), we get In other word, we consider right hand side of (2.15) as follows: Equating I 3 and I 4 , we get the following theorem Now, we consider (2.8) and (2.1) by means of q-Volkenborn integral. Then, by (2.8), we see On the other hand, Therefore, we get the following theorem Theorem 6. For m, k ∈ N, we have By using fermionic p-adic q-integral on Z p left hand side of (2.15), we get Also, we consider right hand side of (2.15) as follows: