- 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.

## 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

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