/ / Article

Research Article | Open Access

Volume 2008 |Article ID 296159 | https://doi.org/10.1155/2008/296159

Young-Hee Kim, Wonjoo Kim, Lee-Chae Jang, "On the -Extension of Apostol-Euler Numbers and Polynomials", Abstract and Applied Analysis, vol. 2008, Article ID 296159, 10 pages, 2008. https://doi.org/10.1155/2008/296159

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

Accepted21 Nov 2008
Published04 Jan 2009

#### Abstract

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

#### 1. Introduction

Let be a fixed odd prime. Throughout this paper, and and will, respectively, denote the ring of -adic rational integers, the field of -adic rational numbers, the complex number field, and the completion of algebraic closure of . Let be the set of natural numbers and . Let be the normalized exponential valuation of with When one talks of -extension, is variously considered as an indeterminate, a complex number or a -adic number . If one normally assumes If then one assumes We also use the notations

For a fixed odd positive integer with , letwhere lies in . The distribution is defined by

We say that is a uniformly differentiable function at a point and denote this property by , if the difference quotients have a limit as . For , the -adic invariant -integral is defined asThe fermionic -adic -measures on are defined asand the fermionic -adic invariant -integral on is defined asfor . For details see .

Classical Euler numbers are defined by the generating functionand these numbers are interpolated by the Euler zeta function which is defined as

After Carlitz  gave -extensions of the classical Bernoulli numbers and polynomials, the -extensions of Bernoulli and Euler numbers and polynomials have been studied by several authors (cf. [116, 1826, 3439]).

By using -adic -integral, the -Euler numbers are defined asThe -Euler numbers are defined by means of the generating function(cf. [8, 26]). Kim  gave a new construction of the -Euler numbers which can be uniquely determined bywith the usual convention of replacing by .

The twisted -Euler numbers and -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 -generalized Euler numbers and polynomials and twisted -generalized Euler numbers and polynomials. Recently, Y. H. Kim et al.  gave the twisted -Euler zeta function associated with twisted -Euler numbers and obtained -Euler's identity. They also have a -extension of the Euler zeta function for negative integers and the -analog of twisted Euler zeta function. Kim  defined twisted -Euler numbers and polynomials of higher order and studied multiple twisted -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, -extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials and numbers have been studied by many authors with great interest. In , Cenkci and Can introduced and investigated -extensions of the Bernoulli polynomials. Choi et al.  have studied some -extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials of order and multiple Hurwitz zeta function.

In this paper, we define Apostol's type -Euler numbers and -Euler polynomials. Then, we have the generating functions of Apostol's type -Euler numbers and -Euler polynomials and the distribution relation for Apostol's type -Euler polynomials. In Section 2, we define Apostol's type -Euler numbers and -Euler polynomials . Then, we obtain the generating functions of and , respectively. We also have the distribution relation for Apostol's type -Euler polynomials. In Section 3, we obtain -zeta function associated with Apostol's type -Euler numbers and Hurwitz's type -zeta function associated with Apostol's type -Euler polynomials for negative integers.

#### 2. on The -Extensions of The Apostol-Euler Numbers and Polynomials

In this section, we will assume with For , let be the cyclic group of order , and let be the space of locally constant space, that is,

Let . We define Apostol's type -Euler numbers byThen, we havewhere are the binomial coefficients.

Apostol's type -Euler polynomials are defined asSincewe have from (2.4) thatBy (2.2) and (2.6), we haveSincewe haveTherefore, we also have

Note that (2.7) and (2.10) are two representations for . Hence, we have the following result.

Theorem 2.1. For and , one has

Now, we will find the generating function of and , respectively. Let be the generating function of . Then, we haveTherefore, the generating function of equalsNote thatFor the generating function of , we haveHence, we obtain the following theorem.

Theorem 2.2. For , one has

Since (2.16) equals to the generating functions (2.17) equals to the generating functions , we have the following result.

Corollary 2.3. For and , one has

Now, we will find the distribution relation for . By (2.4), we haveNote that for odd numbers and ,By (2.19), we haveTherefore, we obtain the distribution relation for as follows.

Theorem 2.4. For , and with , one has

#### 3. Further Remark on The Basic -Zeta Functions Associated with Apostol's Type -Euler Numbers and Polynomials

In this section, we assume that with . Let . For , -zeta function associated with Apostol's type -Euler numbers is defined aswhich is analytic in whole complex -plane. Substituting with into and using Corollary 2.3, then we arrive at

Now, we also consider Hurwitz's type -zeta function associated with the Apostol's type -Euler polynomials as follows:Substituting with into and using Corollary 2.3, then we arrive atHence, we obtain -zeta function associated with Apostol's type -Euler numbers and Hurwitz's type -zeta function associated with Apostol's type -Euler polynomials for negative integers.

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