- Kyung-Won Hwang
^{1}, - Young-Hee Kim
^{2}Email author and - Taekyun Kim
^{2}Email author

**2009**:451217

**DOI: **10.1155/2009/451217

© Kyung-Won Hwang et al. 2009

**Received: **16 May 2009

**Accepted: **25 July 2009

**Published: **23 August 2009

## Abstract

The main purpose of this paper is to present new -extensions of Apostol's type Euler polynomials using the fermionic -adic integral on . We define the - -Euler polynomials and obtain the interpolation functions and the Hurwitz type zeta functions of these polynomials. We define -extensions of Apostol type's Euler polynomials of higher order using the multivariate fermionic -adic integral on . We have the interpolation functions of these - -Euler polynomials. We also give -extensions of Apostol's type Euler polynomials of higher order and have the multiple Hurwitz type zeta functions of these - -Euler polynomials.

## 1. Introduction, Definitions, and Notations

After Carlitz [1] gave -extensions of the classical Bernoulli numbers and polynomials, the -extensions of Bernoulli and Euler numbers and polynomials have been studied by several authors. Many authors have studied on various kinds of -analogues of the Euler numbers and polynomials (cf., [1–39]).T Kim [7–23] has published remarkable research results for -extensions of the Euler numbers and polynomials and their interpolation functions. In [13], T Kim presented a systematic study of some families of multiple -Euler numbers and polynomials. By using the -Volkenborn integration on , he constructed the -adic -Euler numbers and polynomials of higher order and gave the generating function of these numbers and the Euler - -function. In [20], Kim studied some families of multiple -Genocchi and -Euler numbers using the multivariate -adic -Volkenborn integral on , and gave interesting identities related to these numbers. Recently, Kim [21] studied some families of -Euler numbers and polynomials of Nölund's type using multivariate fermionic -adic integral on .

Many authors have studied the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials, and their -extensions (cf., [1, 6, 25, 27, 28, 33–41]). Choi et al. [6] studied some -extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials of order , and multiple Hurwitz zeta function. In [24], Kim et al. defined Apostol's type -Euler numbers and polynomials using the fermionic -adic -integral and obtained the generating functions of these numbers and polynomials, respectively. They also had the distribution relation for Apostol's type -Euler polynomials and obtained -zeta function associated with Apostol's type -Euler numbers and Hurwitz type -zeta function associated with Apostol's type -Euler polynomials for negative integers.

In this paper, we will present new -extensions of Apostol's type Euler polynomials using the fermionic -adic integral on , and then we give interpolation functions and the Hurwitz type zeta functions of these polynomials. We also give -extensions of Apostol's type Euler polynomials of higher order using the multivariate fermionic -adic integral on .

Let be a fixed odd prime number. Throughout this paper , , 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

Now we recall some -notations. The -basic natural numbers are defined by and the -factorial by . The -binomial coefficients are defined by

In Section 2, we define new -extensions of Apostol's type Euler polynomials using the fermionic -adic integral on which will be called the - -Euler polynomials . Then we obtain the interpolation functions and the Hurwitz type zeta functions of these polynomials. In Section 3, we define -extensions of Apostol's type Euler polynomials of higher order using the multivariate fermionic -adic integral on . We have the interpolation functions of these higher-order - -Euler polynomials. In Section 4, we also give -extensions of Apostol's type Euler polynomials of higher order and have the multiple Euler zeta functions of these - -Euler polynomials.

## 2. -Extensions of Apostol's Type Euler Polynomials

Thus, we have the following theorem.

Theorem 2.1.

From (2.14) and (2.15), we obtain the following theorem.

Theorem 2.2.

## 3. -Extensions of Apostol's Type Euler Polynomials of Higher Order

In this section, we give the -extension of Apostol's type Euler polynomials of higher order using the multivariate fermionic -adic integral.

Thus we have the following theorem.

Theorem 3.1.

## 4. -Extension of Apostol's Type Euler Polynomials of Higher Order

In this section, we give the -extension of - -Euler polynomials of higher order using the multivariate fermionic -adic integral.

Note that are called the - -Euler numbers.

When , the - -Euler polynomials are

where is the Gaussian binomial coefficient. From (4.2), we obtain the following theorem.

Theorem 4.1.

## Declarations

### Acknowledgment

The present research has been conducted by the research grant of the Kwangwoon University in 2009.

## Authors’ Affiliations

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