- Research Article
- Open Access

# New Approach to -Euler Numbers and Polynomials

- Taekyun Kim
^{1}, - Lee-Chae Jang
^{2}, - Young-Hee Kim
^{1}Email author and - Seog-Hoon Rim
^{3}

**2010**:431436

https://doi.org/10.1155/2010/431436

© Taekyun Kim et al. 2010

**Received:**11 January 2010**Accepted:**14 March 2010**Published:**30 March 2010

## Abstract

We give a new construction of the -extensions of Euler numbers and polynomials. We present new generating functions which are related to the -Euler numbers and polynomials. We also consider the generalized -Euler polynomials attached to Dirichlet's character and have the generating functions of them. We obtain distribution relations for the -Euler polynomials and have some identities involving -Euler numbers and polynomials. Finally, we derive the -extensions of zeta functions from the Mellin transformation of these generating functions, which interpolate the -Euler polynomials at negative integers.

## Keywords

- Differential Equation
- Partial Differential Equation
- Ordinary Differential Equation
- Mathematical Physic
- Functional Equation

## 1. Introduction

Let be the complex number field. We assume that with and that the -number is defined by in this paper.

Recently, many mathematicians have studied for -Euler and -Bernoulli polynomials and numbers (see [1–18]). Specially, there are papers for the -extensions of Euler polynomials and numbers approaching with two kinds of viewpoint among remarkable papers (see [7, 10]). It is known that the Euler polynomials are defined by , for , and are called the th Euler numbers. The recurrence formula for the original Euler numbers is as follows:

see [7, 10]. As for the -extension of the recurrence formula for the Euler numbers, Kim [10] had the following recurrence formula:

with the usual convention of replacing by . Many researchers have made a wider and deeper study of the -number up to recently (see [1–18]). In the field of number theory and mathematical physics, zeta functions and -functions interpolating these numbers in negative integers have been studied by Cenkci and Can [3], Kim [4–12], and Ozden et al. [16–18].

This research for -Euler numbers seems to be motivated by Carlitz who had constructed the -Bernoulli numbers and polynomials for the first time. In [1, 2], Carlitz considered the recurrence formulae for the -extension of the Bernoulli numbers as follows:

with the usual convention of replacing by . These numbers diverge when , and so Carlitz modified and constructed them as following:

with the usual convention of replacing by . From this, it was shown that . Here are the Bernoulli numbers.

Lately, Carlitz's -Bernoulli numbers have been studied actively by many mathematicians in the field of number theory, discrete mathematics, analysis, mathematical physics, and so on (see [3–18]).

The purpose of this paper is to give a new construction of the -extensions of Euler numbers and polynomials. It is expected that new constructed -Euler numbers and polynomials in this paper are more useful to be applied to various areas related to number theory. In this paper, we present new generating functions which are related to -Euler numbers and polynomials. We also consider the generalized -Euler polynomials attached to Dirichlet's character with an odd conductor and have the generating functions of them. We obtain distribution relations for the -Euler polynomials, and have some identities involving the -Euler numbers and polynomials. Finally, we derive the -extensions of zeta functions from the Mellin transformation of these generating functions. Using the Cauchy residue theorem and Laurent series, we show that these -extensions of zeta functions interpolate the -Euler polynomials at negative integers.

## 2. New Approach to -Euler Numbers and Polynomials

Let be the set of natural numbers and . For with , let us define the -Euler polynomials as follows:

Note that

where are called the th Euler polynomials. In the special case , are called the th -Euler numbers. That is,

From (2.1) and (2.3), we note that

From (2.1) and (2.3), we can easily derive the following equation:

By (2.4) and (2.5), we see that and

Therefore, we obtain the following theorem.

Theorem 2.1.

with the usual convention of replacing by .

Theorem 2.1 of this paper seems to be more interesting and valuable than the -Euler numbers which are introduced in [7, 10].

From (2.1), we note that

Therefore, we obtain the following theorem.

Theorem 2.2.

By (2.1), we see that

By (2.1) and (2.10), we obtain the following theorem.

Theorem 2.3.

From (2.1), we can derive that, for with ,

By (2.12), we see that, for with ,

Therefore, we obtain the following theorem.

Theorem 2.4 (Distribution relation for ).

By (2.1), we observe the following equations:

By (2.15), we obtain the following result.

Theorem 2.5.

where .

Let be Dirichlet's character with an odd conductor . Then we define the generalized -Euler polynomials attached to as follows:

In the special case , are called the th generalized -Euler numbers attached to . Thus the generating functions of the generalized -Euler numbers attached to are as follows:

By (2.1) and (2.17), we see that

Therefore, we obtain the following theorem.

Theorem 2.6.

By (2.17) and (2.18), we see that

Hence

From (2.17), we note that

From (2.17) and (2.23), we have

In (2.19), it is easy to show that

where are called the th generalized Euler polynomials attached to .

For , we now consider the Mellin transformation for the generating function of . That is,

for , and

From (2.26), we define the zeta function as follows:

Note that is analytic function in whole complex -plane. Using the Laurent series and the Cauchy residue theorem, we have

By the same method, we can also obtain the following equation:

For , we define Dirichlet type - -function as

Remark 2.7.

see [19, Lemma ].

## Declarations

### Acknowledgment

The present research has been conducted by the Research Grant of Kwangwoon University in 2010.

## Authors’ Affiliations

## References

- Carlitz L:
**-Bernoulli numbers and polynomials.***Duke Mathematical Journal*1948,**15:**987–1000. 10.1215/S0012-7094-48-01588-9MathSciNetView ArticleMATHGoogle Scholar - Carlitz L:
**-Bernoulli and Eulerian numbers.***Transactions of the American Mathematical Society*1954,**76:**332–350.MathSciNetMATHGoogle Scholar - Cenkci M, Can M:
**Some results on****-analogue of the Lerch zeta function.***Advanced Studies in Contemporary Mathematics*2006,**12**(2):213–223.MathSciNetMATHGoogle Scholar - Kim T:
**On a****-analogue of the****-adic log gamma functions and related integrals.***Journal of Number Theory*1999,**76**(2):320–329. 10.1006/jnth.1999.2373MathSciNetView ArticleMATHGoogle Scholar - Kim T:
**-generalized Euler numbers and polynomials.***Russian Journal of Mathematical Physics*2006,**13**(3):293–298. 10.1134/S1061920806030058MathSciNetView ArticleMATHGoogle Scholar - Kim T:
**-Euler numbers and polynomials associated with****-adic****-integrals.***Journal of Nonlinear Mathematical Physics*2007,**14**(1):15–27. 10.2991/jnmp.2007.14.1.3MathSciNetView ArticleMATHGoogle Scholar - Kim T:
**On the****-extension of Euler and Genocchi numbers.***Journal of Mathematical Analysis and Applications*2007,**326**(2):1458–1465. 10.1016/j.jmaa.2006.03.037MathSciNetView ArticleMATHGoogle Scholar - Kim T:
**-extension of the Euler formula and trigonometric functions.***Russian Journal of Mathematical Physics*2007,**14**(3):275–278. 10.1134/S1061920807030041MathSciNetView ArticleMATHGoogle Scholar - Kim T:
**On the multiple****-Genocchi and Euler numbers.***Russian Journal of Mathematical Physics*2008,**15**(4):481–486. 10.1134/S1061920808040055MathSciNetView ArticleMATHGoogle Scholar - Kim T:
**The modified**-**Euler numbers and polynomials.***Advanced Studies in Contemporary Mathematics*2008,**16**(2):161–170.MathSciNetMATHGoogle Scholar - Kim T:
**Note on the Euler****-zeta functions.***Journal of Number Theory*2009,**129**(7):1798–1804. 10.1016/j.jnt.2008.10.007MathSciNetView ArticleMATHGoogle Scholar - Kim T:
**A note on the generalized****-Euler numbers.***Proceedings of the Jangjeon Mathematical Society*2009,**12**(1):45–50.MathSciNetMATHGoogle Scholar - Kim Y-H, Hwang K-W:
**Symmetry of power sum and twisted Bernoulli polynomials.***Advanced Studies in Contemporary Mathematics*2009,**18**(2):127–133.MathSciNetMATHGoogle Scholar - Kim Y-H, Kim W, Jang L-C:
**On the****-extension of Apostol-Euler numbers and polynomials.***Abstract and Applied Analysis*2008,**2008:**-10.Google Scholar - Kim Y-H, Kim W, Ryoo CS:
**On the twisted****-Euler zeta function associated with twisted****-Euler numbers.***Proceedings of the Jangjeon Mathematical Society*2009,**12**(1):93–100.MathSciNetMATHGoogle Scholar - Ozden H, Cangul IN, Simsek Y:
**Remarks on****-Bernoulli numbers associated with Daehee numbers.***Advanced Studies in Contemporary Mathematics*2009,**18**(1):41–48.MathSciNetGoogle Scholar - Ozden H, Simsek Y:
**A new extension of****-Euler numbers and polynomials related to their interpolation functions.***Applied Mathematics Letters*2008,**21**(9):934–939. 10.1016/j.aml.2007.10.005MathSciNetView ArticleGoogle Scholar - Ozden H, Simsek Y, Rim S-H, Cangul IN:
**A note on****-adic****-Euler measure.***Advanced Studies in Contemporary Mathematics*2007,**14**(2):233–239.MathSciNetGoogle Scholar - Kim T:
**Some identities on the****-Euler polynomials of higher order and****-stirling numbers by the fermionic****-adic integral on**.*Russian Journal of Mathematical Physics*2009,**16:**484–491. 10.1134/S1061920809040037MathSciNetView ArticleMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.