Open Access

A Note on the -Euler Measures

Advances in Difference Equations20092009:956910

DOI: 10.1155/2009/956910

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 [137]):

(11)

The ordinary Euler numbers are defined as (see [137])

(12)

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

(13)

with the usual convention of replacing by

Remark 1.1.

The second kind Euler numbers are also defined as follows (see [25]):
(14)
The Euler polynomials are also defined by
(15)
Thus, we have
(16)

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

(17)
where must be replaced by , symbolically. The -Euler polynomials are given by that is,
(18)

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

(19)

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

(21)

where lies in , (see [137]).

Theorem 2.1.

Let be given by
(22)

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
(23)
By (1.9) and (2.2), we see that
(24)

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:

(25)
The locally constant function on can be integrated by the -adic bounded -Euler measure as follows:
(26)

Therefore, we obtain the following theorem.

Theorem 2.2.

Let be the Dirichlet character with conductor with . Then one has
(27)
Let . From (2.2), we note that
(28)
Thus, we have
(29)

Therefore, we obtain the following theorem and corollary.

Theorem 2.3.

For , one has
(210)

Corollary 2.4.

For , one has
(211)

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

(31)

Thus, we have

(32)
Since for , we have Let . Then we have
(33)

Therefore, we obtain the following theorem.

Theorem 3.1.

Let . Then one has
(34)

Declarations

Acknowledgments

This paper was supported by Jangjeon Mathematical Society.

Authors’ Affiliations

(1)
Division of General Education-Mathematics, Kwangwoon University
(2)
General Education Department, Kookmin University
(3)
Department of Wireless Communications Engineering, Kwangwoon University

References

  1. Cenkci M:The -adic generalized twisted -Euler- -function and its applications. Advanced Studies in Contemporary Mathematics 2007,15(1):37-47.MathSciNetMATHGoogle Scholar
  2. 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
  3. 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
  4. 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
  5. Kim T:On the multiple -Genocchi and Euler numbers. Russian Journal of Mathematical Physics 2008,15(4):481-486. 10.1134/S1061920808040055MathSciNetView ArticleMATHGoogle Scholar
  6. 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
  7. 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
  8. 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
  9. 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
  10. 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
  11. Kim T:Note on -Genocchi numbers and polynomials. Advanced Studies in Contemporary Mathematics 2008,17(1):9-15.MathSciNetMATHGoogle Scholar
  12. Kim T:The modified -Euler numbers and polynomials. Advanced Studies in Contemporary Mathematics 2008,16(2):161-170.MathSciNetMATHGoogle Scholar
  13. 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
  14. Kim T: -Volkenborn integration. Russian Journal of Mathematical Physics 2002,9(3):288-299.MathSciNetMATHGoogle Scholar
  15. Kim T: -Bernoulli numbers and polynomials associated with Gaussian binomial coefficients. Russian Journal of Mathematical Physics 2008,15(1):51-57.MathSciNetView ArticleMATHGoogle Scholar
  16. 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
  17. Kim T:On the von Staudt-Clausen's Theorem for the -Euler numbers. Russian Journal of Mathematical Physics 2009.,16(3):Google Scholar
  18. Kim T: -generalized Euler numbers and polynomials. Russian Journal of Mathematical Physics 2006,13(3):293-298. 10.1134/S1061920806030058MathSciNetView ArticleMATHGoogle Scholar
  19. Kim T:Multiple -adic -function. Russian Journal of Mathematical Physics 2006,13(2):151-157. 10.1134/S1061920806020038MathSciNetView ArticleMATHGoogle Scholar
  20. 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
  21. 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
  22. Kim T: On Euler-Barnes multiple zeta functions. Russian Journal of Mathematical Physics 2003,10(3):261-267.MathSciNetMATHGoogle Scholar
  23. 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
  24. Kim T:Non-Archimedean -integrals associated with multiple Changhee -Bernoulli polynomials. Russian Journal of Mathematical Physics 2003,10(1):91-98.MathSciNetMATHGoogle Scholar
  25. Kim T: Euler numbers and polynomials associated with zeta functions. Abstract and Applied Analysis 2008, 2008:-11.Google Scholar
  26. 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
  27. Schork M:Ward's "calculus of sequences", -calculus and the limit . Advanced Studies in Contemporary Mathematics 2006,13(2):131-141.MathSciNetMATHGoogle Scholar
  28. Simsek Y:Theorems on twisted -function and twisted Bernoulli numbers. Advanced Studies in Contemporary Mathematics 2005,11(2):205-218.MathSciNetMATHGoogle Scholar
  29. 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
  30. 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
  31. 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
  32. 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
  33. 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
  34. 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
  35. 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
  36. 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
  37. 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

© Taekyun Kim et al. 2009

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