On the q-Extension of Apostol-Euler Numbers and Polynomials

Recently, Choi et al. 2008 have studied the q-extensions of the Apostol-Bernoulli and the ApostolEuler polynomials of order n and multiple Hurwitz zeta function. In this paper, we define Apostol’s type q-Euler numbers En,q,ξ and q-Euler polynomials En,q,ξ x . We obtain the generating functions of En,q,ξ and En,q,ξ x , respectively. We also have the distribution relation for Apostol’s type q-Euler polynomials. Finally, we obtain q-zeta function associated with Apostol’s type q-Euler numbers and Hurwitz’s type q-zeta function associated with Apostol’s type q-Euler polynomials for negative integers.


Introduction
Let p be a fixed odd prime. Throughout this paper, Z p , Q p and C and C p will, respectively, denote the ring of p-adic rational integers, the field of p-adic rational numbers, the complex number field, and the completion of algebraic closure of Q p . Let N be the set of natural numbers and Z N ∪ {0}. Let v p be the normalized exponential valuation of C p with |p| p p −v p p p −1 . When one talks of q-extension, q is variously considered as an indeterminate, a complex number q ∈ C, or a p-adic number q ∈ C p . If q ∈ C, one normally assumes |q| < 1. If q ∈ C p , then one assumes |q − 1| p < 1. We also use the notations For a fixed odd positive integer d with p, d 1, let Abstract and Applied Analysis X * 0<a<dp a,p 1 a dpZ p , a dp N Z p {x ∈ X | x ≡ a mod dp N }, where a ∈ Z lies in 0 ≤ a < dp N . The distribution is defined by μ q a dp N Z p q a dp N q . 1.3 We say that f is a uniformly differentiable function at a point a ∈ Z p and denote this property by f ∈ UD Z p , if the difference quotients F f x, y f x − f y / x − y have a limit l f a as x, y → a, a . For f ∈ UD Z p , the p-adic invariant q-integral is defined as The fermionic p-adic q-measures on Z p are defined as μ −q a dp N Z p −q a dp N −q , 1.5 and the fermionic p-adic invariant q-integral on Z p is defined as By using p-adic q-integral, the q-Euler numbers E n,q are defined as The q-Euler numbers E n,q are defined by means of the generating function with the usual convention of replacing E n by E n,q . The twisted q-Euler numbers and q-Euler polynomials are very important in several fields of mathematics and physics, and so they have been studied by many authors. Simsek 37,38 constructed generating functions of q-generalized Euler numbers and polynomials and twisted q-generalized Euler numbers and polynomials. Recently, Y. H. Kim et al. 27 gave the twisted q-Euler zeta function associated with twisted q-Euler numbers and obtained q-Euler's identity. They also have a q-extension of the Euler zeta function for negative integers and the q-analog of twisted Euler zeta function. Kim 24 defined twisted q-Euler numbers and polynomials of higher order and studied multiple twisted q-Euler zeta functions.
The Apostol-Bernoulli and the Apostol-Euler polynomials and numbers have been studied by several authors cf. 15, 17, 32, 33, 40, 41 . Recently, q-extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials and numbers have been studied by many authors with great interest. In 15 , Cenkci and Can introduced and investigated qextensions of the Bernoulli polynomials. Choi et al. 16 have studied some q-extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials of order n and multiple Hurwitz zeta function.
In this paper, we define Apostol's type q-Euler numbers and q-Euler polynomials. Then, we have the generating functions of Apostol's type q-Euler numbers and q-Euler polynomials and the distribution relation for Apostol's type q-Euler polynomials. In Section 2, we define Apostol's type q-Euler numbers E n,q,ξ and q-Euler polynomials E n,q,ξ x . Then, we obtain the generating functions of E n,q,ξ and E n,q,ξ x , respectively. We also have the distribution relation for Apostol's type q-Euler polynomials. In Section 3, we obtain q-zeta function associated with Apostol's type q-Euler numbers and Hurwitz's type q-zeta function associated with Apostol's type q-Euler polynomials for negative integers.

On the q-extensions of the Apostol-Euler numbers and polynomials
In this section, we will assume q ∈ C p with |q − 1| p < 1. For n ∈ Z , let C p n {ξ | ξ p n 1} be the cyclic group of order p n , and let T p be the space of locally constant space, that is, Let ξ ∈ T p . We define Apostol's type q-Euler numbers by Then, we have Therefore, we also have Note that 2.7 and 2.10 are two representations for E n,q,ξ x . Hence, we have the following result. n l x n−l q q lx E l,q,ξ .

2.11
Now, we will find the generating function of E n,q,ξ and E n,q,ξ x , respectively. Let F t be the generating function of E n,q,ξ . Then, we have

2.12
Therefore, the generating function F t of E n,q,ξ equals Note that

2.14
For the generating function of E n,q,ξ x , we have

2.15
Hence, we obtain the following theorem.

2.18
Young-Hee Kim et al.

7
Now, we will find the distribution relation for E n,q,ξ x . By 2.4 , we have y 0 ξ a dy −1 a dy x a dy n q .

2.19
Note that for odd numbers d and p,

2.21
Therefore, we obtain the distribution relation for E n,q,ξ x as follows.

Further remark on the basic q-zeta functions associated with Apostol's type q-Euler numbers and polynomials
In this section, we assume that q ∈ C with |q| < 1. Let ξ ∈ T p . For s ∈ C, q-zeta function associated with Apostol's type q-Euler numbers is defined as ζ q,ξ s 2 q ∞ n 1 ξ n −1 n n s q , 3.1 which is analytic in whole complex s-plane. Substituting s −k with k ∈ Z into ζ q,ξ s and using Corollary 2.3, then we arrive at ζ q,ξ −k 2 q ∞ n 1 ξ n −1 n n k q E k,q,ξ .

3.2
Now, we also consider Hurwitz's type q-zeta function associated with the Apostol's type q-Euler polynomials as follows: ζ q,ξ s, x 2 q ∞ n 0 ξ n −1 n n x s q .

3.3
Substituting s −k with k ∈ Z into ζ q,ξ s, x and using Corollary 2.3, then we arrive at ζ q,ξ −k, x 2 q ∞ n 0 ξ n −1 n n x k q E k,q,ξ x .

3.4
Hence, we obtain q-zeta function associated with Apostol's type q-Euler numbers and Hurwitz's type q-zeta function associated with Apostol's type q-Euler polynomials for negative integers.