Open Access

On the Symmetric Properties for the Generalized Twisted Bernoulli Polynomials

Journal of Inequalities and Applications20092009:164743

DOI: 10.1155/2009/164743

Received: 6 July 2009

Accepted: 18 October 2009

Published: 19 October 2009

Abstract

We study the symmetry for the generalized twisted Bernoulli polynomials and numbers. We give some interesting identities of the power sums and the generalized twisted Bernoulli polynomials using the symmetric properties for the -adic invariant integral.

1. Introduction

Let be a fixed prime number. Throughout this paper, the symbols , , , and denote the ring of rational integers, the ring of -adic integers, the field of -adic rational numbers, and the completion of algebraic closure of , respectively. Let be the set of natural numbers and . Let be the normalized exponential valuation of with .

Let be the space of uniformly differentiable function on . For , the -adic invariant integral on is defined as

(1.1)

(see [1]). From (1.1), we note that

(1.2)

where and . For , let . Then we can derive the following equation from (1.2):

(1.3)

(see [17]).

Let be a fixed positive integer. For , let

(1.4)

where lies in . It is easy to see that

(1.5)

The ordinary Bernoulli polynomials are defined as

(1.6)

and the Bernoulli numbers are defined as (see [119]).

For , let be the -adic locally constant space defined by

(1.7)

where is the cyclic group of order . It is well known that the twisted Bernoulli polynomials are defined as

(1.8)

and the twisted Bernoulli numbers are defined as (see [1518]).

Let be Dirichlet's character with conductor . Then the generalized twisted Bernoulli polynomials attached to are defined as follows:

(1.9)

The generalized twisted Bernoulli numbers attached to , , are defined as (see [16]).

Recently, many authors have studied the symmetric properties of the -adic invariant integrals on , which gave some interesting identities for the Bernoulli and the Euler polynomials (cf. [3, 6, 7, 13, 14, 2027]). The authors of this paper have established various identities by the symmetric properties of the -adic invariant integrals and investigated interesting relationships between the power sums and the Bernoulli polynomials (see [2, 3, 6, 7, 13]).

The twisted Bernoulli polynomials and numbers and the twisted Euler polynomials and numbers are very important in several fields of mathematics and physics(cf. [1518]). The second author has been interested in the twisted Euler numbers and polynomials and the twisted Bernoulli polynomials and studied the symmetry of power sum and twisted Bernoulli polynomials (see [1113]).

The purpose of this paper is to study the symmetry for the generalized twisted Bernoulli polynomials and numbers attached to . In Section 2, we give interesting identities for the power sums and the generalized twisted Bernoulli polynomials using the symmetric properties for the -adic invariant integral.

2. Symmetry for the Generalized Twisted Bernoulli Polynomials

Let be Dirichlet's character with conductor . For , we have

(2.1)

where are the th generalized twisted Bernoulli numbers attached to . We also see that the generalized twisted Bernoulli polynomials attached to are given by

(2.2)

By (2.1) and (2.2), we see that

(2.3)

From (2.3), we derive that

(2.4)

By (1.5) and (2.3), we see that

(2.5)

From (2.2) and (2.5), we obtain that

(2.6)

Thus we have the following theorem from (2.1) and (2.6).

Theorem 2.1.

For , one has
(2.7)

By (1.3) and (1.5), we have that for ,

(2.8)

where . Taking in (2.8), it follows that

(2.9)

Thus, we have

(2.10)

For , let us define the -adic functional as follows:

(2.11)

By (2.10) and (2.11), we see that for ,

(2.12)

From (2.3) and (2.12), we have the following result.

Theorem 2.2.

For and , one has
(2.13)

Let . Then we have that

(2.14)

By (2.9), (2.10), and (2.11), we see that

(2.15)

Now let us define the -adic functional as follows:

(2.16)

Then it follows from (2.14) that

(2.17)

By (2.15) and (2.16), we obtain that

(2.18)

On the other hand, the symmetric property of shows that

(2.19)

Comparing the coefficients on the both sides of (2.18) and (2.19), we have the following theorem.

Theorem 2.3.

Let and . Then one has
(2.20)

We also derive some identities for the generalized twisted Bernoulli numbers. Taking in Theorem 2.3, we have the following corollary.

Corollary 2.4.

Let and . Then one has
(2.21)

Now we will derive another identities for the generalized twisted Bernoulli polynomials using the symmetric property of . From (1.2), (2.15) and (2.17), we see that

(2.22)

From the symmetric property of , we also see that

(2.23)

Comparing the coefficients on the both sides of (2.22) and (2.23), we obtain the following theorem.

Theorem 2.5.

Let and . Then one has
(2.24)

If we take in Theorem 2.5, we also derive the interesting identity for the generalized twisted Bernoulli numbers as follows: for ,

(2.25)

Declarations

Acknowledgment

The present research has been conducted by the research grant of the Kwangwoon University in 2009.

Authors’ Affiliations

(1)
Division of General Education-Mathematics, Kwangwoon University

References

  1. Kim T: -Volkenborn integration. Russian Journal of Mathematical Physics 2002,9(3):288–299.MathSciNetMATHGoogle Scholar
  2. Kim T: On the symmetries of the -Bernoulli polynomials. Abstract and Applied Analysis 2008, 2008:-7.Google Scholar
  3. 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
  4. Kim T: On the multiple -Genocchi and Euler numbers. Russian Journal of Mathematical Physics 2008,15(4):481–486. 10.1134/S1061920808040055MathSciNetView ArticleMATHGoogle Scholar
  5. Kim T: Note on -Genocchi numbers and polynomials. Advanced Studies in Contemporary Mathematics 2008,17(1):9–15.MathSciNetMATHGoogle Scholar
  6. Kim T: Symmetry of power sum polynomials and multivariate fermionic -adic invariant integral on . Russian Journal of Mathematical Physics 2009,16(1):93–96. 10.1134/S1061920809010063MathSciNetView ArticleMATHGoogle Scholar
  7. Kim T, Rim S-H, Lee B: Some identities of symmetry for the generalized Bernoulli numbers and polynomials. Abstract and Applied Analysis 2009, 2009:-8.Google Scholar
  8. Carlitz L: -Bernoulli numbers and polynomials. Duke Mathematical Journal 1948, 15: 987–1000. 10.1215/S0012-7094-48-01588-9MathSciNetView ArticleMATHGoogle Scholar
  9. 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
  10. Hegazi AS, Mansour M: A note on -Bernoulli numbers and polynomials. Journal of Nonlinear Mathematical Physics 2006,13(1):9–18. 10.2991/jnmp.2006.13.1.2MathSciNetView ArticleMATHGoogle Scholar
  11. Kim Y-H: On the -adic interpolation functions of the generalized twisted -Euler numbers. International Journal of Mathematical Analysis 2008,3(18):897–904.MathSciNetMATHGoogle Scholar
  12. 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
  13. 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
  14. Kupershmidt BA: Reflection symmetries of -Bernoulli polynomials. Journal of Nonlinear Mathematical Physics 2005, 12: 412–422. 10.2991/jnmp.2005.12.s1.34MathSciNetView ArticleGoogle Scholar
  15. Simsek Y: Theorems on twisted -function and twisted Bernoulli numbers. Advanced Studies in Contemporary Mathematics 2005,11(2):205–218.MathSciNetMATHGoogle Scholar
  16. 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
  17. 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
  18. Simsek Y, Kurt V, Kim D: New approach to the complete sum of products of the twisted -Bernoulli numbers and polynomials. Journal of Nonlinear Mathematical Physics 2007,14(1):44–56. 10.2991/jnmp.2007.14.1.5MathSciNetView ArticleMATHGoogle Scholar
  19. 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
  20. 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
  21. Kim T: A note on some formulae for the -Euler numbers and polynomials. Proceedings of the Jangjeon Mathematical Society 2006,9(2):227–232.MathSciNetMATHGoogle Scholar
  22. Kim T: Note on Dedekind type DC sums. Advanced Studies in Contemporary Mathematics 2009,18(2):249–260.MathSciNetMATHGoogle Scholar
  23. Kim T: A note on the generalized -Euler numbers. Proceedings of the Jangjeon Mathematical Society 2009,12(1):45–50.MathSciNetMATHGoogle Scholar
  24. Kim T: New approach to -Euler, Genocchi numbers and their interpolation functions. Advanced Studies in Contemporary Mathematics 2009,18(2):105–112.MathSciNetMATHGoogle Scholar
  25. Kim T: Symmetry identities for the twisted generalized Euler polynomials. Advanced Studies in Contemporary Mathematics 2009,19(2):151–155.MathSciNetMATHGoogle Scholar
  26. 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
  27. Ozden H, Cangul IN, Simsek Y: Remarks on -Bernoulli numbers associated with Daehee numbers. Advanced Studies in Contemporary Mathematics 2009,18(1):41–48.MathSciNetMATHGoogle Scholar

Copyright

© T. Kim and Y.-H. Kim. 2009

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.