- Research Article
- Open Access
- Published:

# A Note on the -Euler Measures

*Advances in Difference Equations*
**volume 2009**, Article number: 956910 (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.

The second kind Euler numbers are also defined as follows (see [25]):

The Euler polynomials are also defined by

Thus, we have

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

where must be replaced by , symbolically. The -Euler polynomials are given by that is,

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.

Let be given by

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

Proof.

It is sufficient to show that

By (1.9) and (2.2), we see that

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:

The locally constant function on can be integrated by the -adic bounded -Euler measure as follows:

Therefore, we obtain the following theorem.

Theorem 2.2.

Let be the Dirichlet character with conductor with . Then one has

Let . From (2.2), we note that

Thus, we have

Therefore, we obtain the following theorem and corollary.

Theorem 2.3.

For , one has

Corollary 2.4.

For , one has

## 3. -adic --Series

In this section, we assume that with . Let denote the Teichmüller character . For , we set . Note that , and is defined by , for . For ,we define

Thus, we have

Since for , we have Let . Then we have

Therefore, we obtain the following theorem.

Theorem 3.1.

Let . Then one has

## References

Cenkci M:

**The**-adic generalized twisted -Euler-**-function and its applications.***Advanced Studies in Contemporary Mathematics*2007,**15**(1):37-47.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.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.Kim T:

**-extension of the Euler formula and trigonometric functions.***Russian Journal of Mathematical Physics*2007,**14**(3):275-278. 10.1134/S1061920807030041Kim T:

**On the multiple****-Genocchi and Euler numbers.***Russian Journal of Mathematical Physics*2008,**15**(4):481-486. 10.1134/S1061920808040055Rim S-H, Kim T:

**A note on**-adic Euler measure on**.***Russian Journal of Mathematical Physics*2006,**13**(3):358-361. 10.1134/S1061920806030113Kim T:-Euler numbers and polynomials associated with -adic

**-integrals.***Journal of Nonlinear Mathematical Physics*2007,**14**(1):15-27. 10.2991/jnmp.2007.14.1.3Leyendekkers 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.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.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.Kim T:

**Note on****-Genocchi numbers and polynomials.***Advanced Studies in Contemporary Mathematics*2008,**17**(1):9-15.Kim T:

**The modified****-Euler numbers and polynomials.***Advanced Studies in Contemporary Mathematics*2008,**16**(2):161-170.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.1999.2373Kim T:

**-Volkenborn integration.***Russian Journal of Mathematical Physics*2002,**9**(3):288-299.Kim T:

**-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients.***Russian Journal of Mathematical Physics*2008,**15**(1):51-57.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/S1061920807020045Kim T:

**On the von Staudt-Clausen's Theorem for the****-Euler numbers.***Russian Journal of Mathematical Physics*2009.,**16**(3):Kim T:

**-generalized Euler numbers and polynomials.***Russian Journal of Mathematical Physics*2006,**13**(3):293-298. 10.1134/S1061920806030058Kim T:

**Multiple**-adic**-function.***Russian Journal of Mathematical Physics*2006,**13**(2):151-157. 10.1134/S1061920806020038Kim 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.Kim T:

**Analytic continuation of multiple****-zeta functions and their values at negative integers.***Russian Journal of Mathematical Physics*2004,**11**(1):71-76.Kim T:

**On Euler-Barnes multiple zeta functions.***Russian Journal of Mathematical Physics*2003,**10**(3):261-267.Kim T:

**Symmetry**-adic invariant integral on**for Bernoulli and Euler polynomials.***Journal of Difference Equations and Applications*2008,**14**(12):1267-1277. 10.1080/10236190801943220Kim T:

**Non-Archimedean**-integrals associated with multiple Changhee**-Bernoulli polynomials.***Russian Journal of Mathematical Physics*2003,**10**(1):91-98.Kim T:

**Euler numbers and polynomials associated with zeta functions.***Abstract and Applied Analysis*2008,**2008:**-11.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.Schork M:

**Ward's "calculus of sequences",**-calculus and the limit**.***Advanced Studies in Contemporary Mathematics*2006,**13**(2):131-141.Simsek Y:

**Theorems on twisted****-function and twisted Bernoulli numbers.***Advanced Studies in Contemporary Mathematics*2005,**11**(2):205-218.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.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.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.Simsek Y:

**On**-adic twisted -**-functions related to generalized twisted Bernoulli numbers.***Russian Journal of Mathematical Physics*2006,**13**(3):340-348. 10.1134/S1061920806030095Ozden H, Simsek Y, Rim S-H, Cangul IN:

**A note on**-adic**-Euler measure.***Advanced Studies in Contemporary Mathematics*2007,**14**(2):233-239.Ozden H, Cangul IN, Simsek Y:

**Multivariate interpolation functions of higher-order****-Euler numbers and their applications.***Abstract and Applied Analysis*2008,**2008:**-16.Tuenter HJH:

**A symmetry of power sum polynomials and Bernoulli numbers.***The American Mathematical Monthly*2001,**108**(3):258-261. 10.2307/2695389Cenkci 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/S106192080804002XAtanassov 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.

## Acknowledgments

This paper was supported by Jangjeon Mathematical Society.

## Author information

### Authors and Affiliations

### Corresponding author

## Rights and permissions

**Open Access**
This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (
https://creativecommons.org/licenses/by/2.0
), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## About this article

### Cite this article

Kim, T., Hwang, KW. & Lee, B. A Note on the -Euler Measures.
*Adv Differ Equ* **2009**, 956910 (2009). https://doi.org/10.1155/2009/956910

Received:

Accepted:

Published:

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

### Keywords

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