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Abstract and Applied Analysis
Volume 2008, Article ID 581582, 11 pages
http://dx.doi.org/10.1155/2008/581582
Research Article

Euler Numbers and Polynomials Associated with Zeta Functions

Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, South Korea

Received 16 January 2008; Accepted 19 April 2008

Academic Editor: Lance Littlejohn

Copyright © 2008 Taekyun Kim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

For , the Euler zeta function and the Hurwitz-type Euler zeta function are defined by , and . Thus, we note that the Euler zeta functions are entire functions in whole complex -plane, and these zeta functions have the values of the Euler numbers or the Euler polynomials at negative integers. That is, , and . We give some interesting identities between the Euler numbers and the zeta functions. Finally, we will give the new values of the Euler zeta function at positive even integers.

1. Introduction

Throughout this paper, , , , , , and will, respectively, denote the ring of rational integers, the field of rational numbers, the field of complex numbers, the ring -adic rational integers, the field of -adic rational numbers, and the completion of the algebraic closure of . Let be the normalized exponential valuation of such that . If , we normally assume that . We use the notation Hence, for any with in the present -adic case.

Let be a fixed odd prime. For a fixed positive integer with , let where lies in .

In [1], we note that is distribution on for with This distribution yields an integral as follows: which has a sense as we see readily that the limit is convergent (see [1]). Let . Then, we have the fermionic -adic integral on as follows: (cf. [15]). For any positive integer we set (cf. [13, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]) and this can be extended to a distribution on . This distribution yields -adic bosonic -integral as follows (see [11, 20]): where with values in , (cf. [2, 11, 1620]). In view of notation, can be written symbolically as If we take , then we can derive the -extension of Bernoulli numbers and polynomials from -adic -integrals on as follows: (cf. [11, 20]). Thus, we note that (cf. [11, 14, 20]). In the complex plane, the ordinary Bernoulli numbers are a sequence of signed rational numbers that can be defined by the identity

(cf. [133]).

These numbers arise in the series expansions of trigonometric functions, and are extremely important in number theory and analysis. From the generating function of Bernoulli numbers, we note that and for . It is well known that Riemann zeta function is defined by We also note that the Riemann zeta function is closely related to Bernoulli numbers at positive integer or negative integer in the complex plane. Riemann did develop the theory of analytic continuation needed to rigorously define for all . From this zeta function, he derived the following formula (cf. [133]): Thus, we note that if is an even integer and greater than 0. These are called the trivial zeros of the zeta function. In 1859, starting with Euler's factorization of the zeta function he derived an explicit formula for the prime numbers in terms of zeros of the zeta function. He also posed the Riemann hypothesis: if then either is a trivial zero or lies on the critical line (cf. [4, 5, 1620, 27, 28, 29, 30, 31, 32, 33]). It is well known that Thus, (cf. [4, 5, 10, 1620, 27, 28, 29, 30, 31, 32, 33]). From this, we can derive the following famous formula.

Lemma 1.1. For ,

It is easy to see that However, it is not known the values of for . In the case of , Apery proved that is irrational number (see [34]). The constants in the Taylor series expansion (cf. [35, 10, 27]) are known as the first-kind Euler numbers. From the generating function of the first-kind Euler numbers, we note that The first few are and for The Euler polynomials are also defined by For , the Euler zeta function and Hurwitz's type Euler zeta function are defined by (cf. [2, 4, 5, 9, 10, 27]). Thus, we note that Euler zeta functions are entire functions in the whole complex -plane and these zeta functions have the values of the Euler numbers or Euler polynomials at negative integers. That is, (cf. [2, 4, 5, 9, 10, 27]).

In this paper, we give some interesting identities between Euler numbers and zeta functions. Finally, we will give the values of the Euler zeta function at positive even integers.

2. Preliminaries/Euler Numbers Associated with -adic Fermionic Integrals

Let be the translation defined by . Then we have If we take , then we can derive the first-kind Euler polynomials from the integral equation of as follows: That is, For , we have the following integral equation: From this we note that

Let (or ). By using the fermionic -adic -integral on , we see that see [12], Thus, we obtain see [12]. From this we note that By the same motivation, we can also observe that see [12]. These formulae are also treated in Section 3.

Let . Then we can derive the generating function of the second-kind Euler numbers from fermionic -adic integral equation as follows: Thus, we have where we have used the symbolic notation for . The first few are , for . In particular, Recently, Simsek, Ozden, Cangül, Cenkci, Kurt, and others have studied the various extensions of the first kind Euler numbers by using fernionic -adic invariant -integral on , see [25, 16, 20, 27]. It seems to be also interesting to study the -extensions of the second-kind Euler numbers due to Simsek et al. (see [4, 5, 16]).

3. Some Relationships between Euler Numbers and Zeta Functions

In this section, we also consider Bernoulli and the second Euler numbers in the complex plane. The second-kind Euler numbers are defined by the following expansion: (cf. [10]). From (1.18) and (3.1), we can derive the following equation: By (3.2) and (1.18), we easily see that and for As Euler formula, it is well known that From (3.3), we note that Thus, we have From (3.4), we derive The Fourier series of an odd function on the interval is the sine series: where Let us consider on . From (3.6) and (3.7), we note that where In (3.8), if we take , then we have From (3.10), we note that In (3.5), it is easy to see that By (3.11) and (3.12), we obtain the following.

Theorem 3.1. For ,

It is easy to see that By (3.13) and (3.14), we obtain the following.

Corollary 3.2. For ,

By simple calculation, we easily see that Thus, we have From (3.17), we can easily derive By (3.3), we also see that Thus, we have By (3.18) and (3.20), we obtain the following.

Theorem 3.3. For ,
where are the first-kind Euler numbers.

It is easy to see that Therefore, we obtain the following corollary.

Corollary 3.4. For ,

Now we try to give the new value of the Euler zeta function at positive integers. From the definition of the Euler zeta function, we note that By (3.24), Theorem 3.3, and Corollary 3.4, we obtain the following theorem.

Theorem 3.5. For ,

Remark 3.6. We note that , , and . For with , , - -function is defined by (cf. [10, 14]). Note that is analytic continuation in with only one simple pole at , and (cf. [14]). By simple calculation, we easily see that where are Carlitz's -Euler numbers with (cf. [6, 21, 22]). If , then we have For , and , it is easy to see that From (3.30) and Theorem 3.1, we can also derive the following equation: Thus, we have

Acknowledgments

The present research has been conducted by the research grant of Kwangwoon University in 2008. The author wishes to express his sincere gratitude to the referees and Professor Lance L. Littlejohn for their valuable suggestions and comments.

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