- Research Article
- Open Access

- Taekyun Kim
^{1}, - Kyung-Won Hwang
^{2}Email author and - Byungje Lee
^{3}

**2009**:956910

https://doi.org/10.1155/2009/956910

© Taekyun Kim et al. 2009

**Received:**6 March 2009**Accepted:**20 May 2009**Published:**28 June 2009

## Abstract

Properties of -extensions of Euler numbers and polynomials which generalize those satisfied by and are used to construct -extensions of -adic Euler measures and define -adic - -series which interpolate -Euler numbers at negative integers. Finally, we give Kummer Congruence for the -extension of ordinary Euler numbers.

## Keywords

- Partial Differential Equation
- Ordinary Differential Equation
- Functional Equation
- Complex Number
- Prime Number

## 1. Introduction

Let be a fixed 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 normalized exponential valuation of with . When one talks of -extension, is variously considered as an indeterminate, a complex number or -adic numbers . If , one normally assumes . If , one normally assumes . In this paper, we use the notations of -number as follows (see [1–37]):

The ordinary Euler numbers are defined as (see [1–37])

where is written as when is replaced by . From the definition of Euler number, we can derive

with the usual convention of replacing by

Remark 1.1.

In [7], -Euler numbers, , can be determined inductively by

Let be a fixedoddpositive integer. Then we have (see [7])

We use (1.9) to get bounded -adic -Euler measures and finally take the Mellin transform to define -adic - -series which interpolate -Euler numbers at negative integers.

## 2. -adic -Euler Measures

Let be a fixed odd positive integer, and let be a fixed odd prime number. Define

Theorem 2.1.

Then extends to a -valued measure on the compact open sets . Note that , where is fermionic measure on (see [7])

Proof.

and we easily see that for some constant .

Let be a Dirichlet character with conductor with . Then we define the generalized -Euler numbers attached to as follows:

Therefore, we obtain the following theorem.

Theorem 2.2.

Therefore, we obtain the following theorem and corollary.

Theorem 2.3.

Corollary 2.4.

## 3. -adic - -Series

## Declarations

### Acknowledgments

This paper was supported by Jangjeon Mathematical Society.

## Authors’ Affiliations

## References

- Cenkci M:
**The**-adic generalized twisted -Euler-**-function and its applications.***Advanced Studies in Contemporary Mathematics*2007,**15**(1):37-47.MathSciNetMATHGoogle Scholar - Cenkci M, Simsek Y, Kurt V:
**Further remarks on multiple**-adic -**-function of two variables.***Advanced Studies in Contemporary Mathematics*2007,**14**(1):49-68.MathSciNetMATHGoogle Scholar - Cenkci M, Can M, Kurt V:
-adic interpolation functions and Kummer-type congruences for
**-twisted Euler numbers.***Advanced Studies in Contemporary Mathematics*2004,**9**(2):203-216.MathSciNetMATHGoogle 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 - Rim S-H, Kim T:
**A note on**-adic Euler measure on**.***Russian Journal of Mathematical Physics*2006,**13**(3):358-361. 10.1134/S1061920806030113MathSciNetView 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 - Leyendekkers JV, Shannon AG, Wong CK:
**Integer structure analysis of the product of adjacent integers and Euler's extension of Fermat's last theorem.***Advanced Studies in Contemporary Mathematics*2008,**17**(2):221-229.MathSciNetMATHGoogle Scholar - Ozden H, Cangul IN, Simsek Y:
**Remarks on sum of products of****-twisted Euler polynomials and numbers.***Journal of Inequalities and Applications*2008,**2008:**-8.Google Scholar - Srivastava HM, Kim T, Simsek Y:
-Bernoulli numbers and polynomials associated with multiple
-zeta functions and basic
**-series.***Russian Journal of Mathematical Physics*2005,**12**(2):241-268.MathSciNetMATHGoogle Scholar - Kim T:
**Note on****-Genocchi numbers and polynomials.***Advanced Studies in Contemporary Mathematics*2008,**17**(1):9-15.MathSciNetMATHGoogle Scholar - Kim T:
**The modified****-Euler numbers and polynomials.***Advanced Studies in Contemporary Mathematics*2008,**16**(2):161-170.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:
**-Volkenborn integration.***Russian Journal of Mathematical Physics*2002,**9**(3):288-299.MathSciNetMATHGoogle Scholar - Kim T:
**-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients.***Russian Journal of Mathematical Physics*2008,**15**(1):51-57.MathSciNetView ArticleMATHGoogle Scholar - Kim T, Choi JY, Sug JY:
**Extended**-Euler numbers and polynomials associated with fermionic -adic -integral on**.***Russian Journal of Mathematical Physics*2007,**14**(2):160-163. 10.1134/S1061920807020045MathSciNetView ArticleMATHGoogle Scholar - Kim T:On the von Staudt-Clausen's Theorem for the -Euler numbers. Russian Journal of Mathematical Physics 2009.,16(3):Google 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:
**Multiple**-adic**-function.***Russian Journal of Mathematical Physics*2006,**13**(2):151-157. 10.1134/S1061920806020038MathSciNetView ArticleMATHGoogle Scholar - Kim T:
**Power series and asymptotic series associated with the**-analog of the two-variable -adic**-function.***Russian Journal of Mathematical Physics*2005,**12**(2):186-196.MathSciNetMATHGoogle Scholar - Kim T:
**Analytic continuation of multiple****-zeta functions and their values at negative integers.***Russian Journal of Mathematical Physics*2004,**11**(1):71-76.MathSciNetMATHGoogle Scholar - Kim T:
**On Euler-Barnes multiple zeta functions.***Russian Journal of Mathematical Physics*2003,**10**(3):261-267.MathSciNetMATHGoogle Scholar - Kim T:
**Symmetry**-adic invariant integral on**for Bernoulli and Euler polynomials.***Journal of Difference Equations and Applications*2008,**14**(12):1267-1277. 10.1080/10236190801943220MathSciNetView ArticleMATHGoogle Scholar - Kim T:
**Non-Archimedean**-integrals associated with multiple Changhee**-Bernoulli polynomials.***Russian Journal of Mathematical Physics*2003,**10**(1):91-98.MathSciNetMATHGoogle Scholar - Kim T:
**Euler numbers and polynomials associated with zeta functions.***Abstract and Applied Analysis*2008,**2008:**-11.Google Scholar - Kim T, Kim Y-H, Hwang K-W:
**On the****-extensions of the Bernoulli and Euler numbers, related identities and Lerch zeta function.***Proceedings of the Jangjeon Mathematical Society*2009,**12:**1-16.MathSciNetGoogle Scholar - Schork M:
**Ward's "calculus of sequences",**-calculus and the limit**.***Advanced Studies in Contemporary Mathematics*2006,**13**(2):131-141.MathSciNetMATHGoogle Scholar - Simsek Y:
**Theorems on twisted****-function and twisted Bernoulli numbers.***Advanced Studies in Contemporary Mathematics*2005,**11**(2):205-218.MathSciNetMATHGoogle Scholar - Simsek Y:
**Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions.***Advanced Studies in Contemporary Mathematics*2008,**16**(2):251-278.MathSciNetMATHGoogle Scholar - Zhang Z, Yang H:
**Some closed formulas for generalized Bernoulli-Euler numbers and polynomials.***Proceedings of the Jangjeon Mathematical Society*2008,**11**(2):191-198.MathSciNetMATHGoogle Scholar - Simsek Y, Yurekli O, Kurt V:
**On interpolation functions of the twisted generalized Frobenius-Euler numbers.***Advanced Studies in Contemporary Mathematics*2007,**15**(2):187-194.MathSciNetMATHGoogle Scholar - Simsek Y:
**On**-adic twisted -**-functions related to generalized twisted Bernoulli numbers.***Russian Journal of Mathematical Physics*2006,**13**(3):340-348. 10.1134/S1061920806030095MathSciNetView ArticleMATHGoogle 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 - Ozden H, Cangul IN, Simsek Y:
**Multivariate interpolation functions of higher-order****-Euler numbers and their applications.***Abstract and Applied Analysis*2008,**2008:**-16.Google Scholar - Tuenter HJH:
**A symmetry of power sum polynomials and Bernoulli numbers.***The American Mathematical Monthly*2001,**108**(3):258-261. 10.2307/2695389MathSciNetView ArticleMATHGoogle Scholar - Cenkci M, Simsek Y, Kurt V:
**Multiple two-variable**-adic - -function and its behavior at**.***Russian Journal of Mathematical Physics*2008,**15**(4):447-459. 10.1134/S106192080804002XMathSciNetView ArticleMATHGoogle Scholar - Atanassov KT, Vassilev-Missana MV:
**On one of Murthy-Ashbacher's conjectures related to Euler's totient function.***Proceedings of the Jangjeon Mathematical Society*2006,**9**(1):47-49.MathSciNetMATHGoogle 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.