- Research Article
- Open Access

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

- Taekyun Kim
^{1}Email author, - Lee-Chae Jang
^{2}, - Young-Hee Kim
^{1}and - Kyung-Won Hwang
^{3}

**2009**:640152

https://doi.org/10.1155/2009/640152

© Taekyun Kim et al. 2009

**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 .

## Keywords

- Positive Integer
- Natural Number
- Prime Number
- Rational Number
- Differentiable Function

## 1. Introduction

and the generalized Bernoulli numbers attached to , , are defined as (see [1–20, 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

(see [25]).

From (2.16), we note that

By the symmetry of in and , we see that

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

Theorem 2.1.

Remark 2.2.

(see [25]).

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

Theorem 2.3.

Remark 2.4.

(see [25]).

## Declarations

### Acknowledgment

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

## 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:
**Multiple two-variable**-**adic**- -**function and its behavior at**.*Russian Journal of Mathematical Physics*2008,**15**(4):447–459. 10.1134/S106192080804002XMathSciNetView ArticleMATHGoogle Scholar - 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 - 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:
**A note on**-**Volkenborn integration.***Proceedings of the Jangjeon Mathematical Society*2005,**8**(1):13–17.MathSciNetMATHGoogle 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 - Kim T:
**A note on**-**adic**-**integral on****associated with**-**Euler numbers.***Advanced Studies in Contemporary Mathematics*2007,**15**(2):133–137.MathSciNetMATHGoogle Scholar - 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.2006.03.037MathSciNetView ArticleMATHGoogle Scholar - Kim T:
**On**-**adic**- -**functions and sums of powers.***Journal of Mathematical Analysis and Applications*2007,**329**(2):1472–1481. 10.1016/j.jmaa.2006.07.071MathSciNetView 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 - Kim T:
**New approach to**-**Euler, Genocchi numbers and their interpolation functions.***Advanced Studies in Contemporary Mathematics*2009,**18**(2):105–112.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: 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
- 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:
**-Volkenborn integration.***Russian Journal of Mathematical Physics*2002,**9**(3):288–299.MathSciNetMATHGoogle Scholar - 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 - Pečarić J, Vukelić A:
**General dual Euler-Simpson formulae.***Journal of Mathematical Inequalities*2008,**2**(4):511–526.MathSciNetMATHGoogle Scholar - Simsek Y:
**Complete sum of products of -extension of the Euler polynomials and numbers.**http://arxiv.org/abs/0707.2849 - 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 - 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 - Kim T: Symmetry properties of higher order Bernoulli polynomials. (communicated) (communicated)Google 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 - 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.MathSciNetMATHGoogle Scholar - 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/S1061920807020033MathSciNetView ArticleMATHGoogle Scholar - 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

## Copyright

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.