## Journal of Inequalities and Applications

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# On the Identities of Symmetry for the Generalized Bernoulli Polynomials Attached to of Higher Order

Journal of Inequalities and Applications20092009:640152

DOI: 10.1155/2009/640152

Received: 5 June 2009

Accepted: 5 August 2009

Published: 26 August 2009

## Abstract

We give some interesting relationships between the power sums and the generalized Bernoulli numbers attached to of higher order using multivariate -adic invariant integral on .

## 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 (see [124]). Let be the space of uniformly differentiable function on . Let be a fixed positive integer. For , let
(1.1)
where lies in . For , the -adic invariant integral on is defined as
(1.2)
(see [1119]). From (1.2), we note that
(1.3)
where and . Let . Then we can derive the following equation from (1.3):
(1.4)
(see [111]). Let be the Dirichlet's character with conductor . Then the generalized Bernoulli polynomials attached to are defined as
(1.5)

and the generalized Bernoulli numbers attached to , , are defined as (see [120, 25]). The purpose of this paper is to derive some identities of symmetry for the generalized Bernoulli polynomials attached to of higher order.

## 2. Symmetric Properties for the Generalized Bernoulli Polynomials of Higher Order

Let be the Dirichlet's character with conductor . Then we note that
(2.1)
where are the th generalized Bernoulli numbers attached to (see [7, 9, 15, 25]). Now we also see that the generalized Bernoulli polynomials attached to are given by
(2.2)
By (2.1) and (2.2), we have
(2.3)
(see [15, 25]), and
(2.4)
(see [119, 25]). For , we obtain that
(2.5)
where . Thus, we have
(2.6)
Then
(2.7)
Let us define the -adic function as follows:
(2.8)
(see [25]). By (2.7) and (2.8), we see that
(2.9)
(see [25]). Thus, we have
(2.10)
This means that
(2.11)

(see [25]).

The generalized Bernoulli polynomials attached to of order , which is denoted by , are defined as
(2.12)
Then the values of at are called the generalized Bernoulli numbers attached to of order . When , the polynomials of numbers are called the generalized Bernoulli polynomials or numbers attached to . Let . Then we set
(2.13)
where
(2.14)
In (2.13), we note that is symmetric in . From (2.13), we derive
(2.15)
It is easy to see that
(2.16)

From (2.16), we note that

(2.17)

By the symmetry of in and , we see that

(2.18)

By comparing the coefficients on the both sides of (2.17) and (2.18), we see the following theorem.

Theorem 2.1.

For , one has
(2.19)

Remark 2.2.

Let and in (1.4). Then we have
(2.20)

(see [25]).

We also calculate that
(2.21)
From the symmetric property of in and , we derive
(2.22)

By comparing the coefficients on the both sides of (2.21) and (2.22), we obtain the following theorem.

Theorem 2.3.

For , one has
(2.23)

Remark 2.4.

Let and in (2.23). We have
(2.24)

(see [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
(2)
Department of Mathematics and Computer Science, Konkook University
(3)
Department of General Education, Kookmin University

## References

1. Cenkci M: The -adic generalized twisted -Euler- -function and its applications. Advanced Studies in Contemporary Mathematics 2007,15(1):37–47.
2. 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/S106192080804002X
3. Jang L-C, Kim S-D, Park D-W, Ro Y-S: A note on Euler number and polynomials. Journal of Inequalities and Applications 2006, 2006:-5.Google Scholar
4. Kim T: -Bernoulli numbers and polynomials associated with Gaussian binomial coefficients. Russian Journal of Mathematical Physics 2008,15(1):51–57.
5. Kim T: A note on -Volkenborn integration. Proceedings of the Jangjeon Mathematical Society 2005,8(1):13–17.
6. Kim T: -Euler numbers and polynomials associated with -adic -integrals. Journal of Nonlinear Mathematical Physics 2007,14(1):15–27. 10.2991/jnmp.2007.14.1.3
7. Kim T: A note on -adic -integral on associated with -Euler numbers. Advanced Studies in Contemporary Mathematics 2007,15(2):133–137.
8. Kim T: On the -extension of Euler and Genocchi numbers. Journal of Mathematical Analysis and Applications 2007,326(2):1458–1465. 10.1016/j.jmaa.2006.03.037
9. Kim T: On -adic - -functions and sums of powers. Journal of Mathematical Analysis and Applications 2007,329(2):1472–1481. 10.1016/j.jmaa.2006.07.071
10. Kim T: On the multiple -Genocchi and Euler numbers. Russian Journal of Mathematical Physics 2008,15(4):481–486. 10.1134/S1061920808040055
11. Kim T: New approach to -Euler, Genocchi numbers and their interpolation functions. Advanced Studies in Contemporary Mathematics 2009,18(2):105–112.
12. 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.1999.2373
13. Kim T: Sums of products of -Euler numbers. to appear in Journal of Computational Analysis and Applications to appear in Journal of Computational Analysis and ApplicationsGoogle Scholar
14. 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/S1061920807020045
15. Kim T: -Volkenborn integration. Russian Journal of Mathematical Physics 2002,9(3):288–299.
16. 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
17. Pečarić J, Vukelić A: General dual Euler-Simpson formulae. Journal of Mathematical Inequalities 2008,2(4):511–526.
18. Simsek Y: Complete sum of products of -extension of the Euler polynomials and numbers. http://arxiv.org/abs/0707.2849
19. Simsek Y: On -adic twisted - - functions related to generalized twisted Bernoulli numbers. Russian Journal of Mathematical Physics 2006,13(3):340–348. 10.1134/S1061920806030095
20. 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.2007.14.1.5
21. Kim T: Symmetry properties of higher order Bernoulli polynomials. (communicated) (communicated)Google Scholar
22. 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.
23. Tekcan A, Özkoç A, Gezer B, Bizim O: Some relations involving the sums of Fibonacci numbers. Proceedings of the Jangjeon Mathematical Society 2008,11(1):1–12.
24. Khrennikov AYu: Generalized probabilities taking values in non-Archimedean fields and in topological groups. Russian Journal of Mathematical Physics 2007,14(2):142–159. 10.1134/S1061920807020033
25. 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