- Research Article
- Open Access
- Published:

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

*Journal of Inequalities and Applications*
**volume 2009**, Article number: 640152 (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 [1–24]). Let be the space of uniformly differentiable function on . Let be a fixed positive integer. For , let

where lies in . For , the -adic invariant integral on is defined as

(see [11–19]). From (1.2), we note that

where and . Let . Then we can derive the following equation from (1.3):

(see [1–11]). Let be the Dirichlet's character with conductor . Then the generalized Bernoulli polynomials attached to are defined as

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

Let be the Dirichlet's character with conductor . Then we note that

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

By (2.1) and (2.2), we have

(see [1–19, 25]). For , we obtain that

where . Thus, we have

Then

Let us define the -adic function as follows:

(see [25]). By (2.7) and (2.8), we see that

(see [25]). Thus, we have

This means that

(see [25]).

The generalized Bernoulli polynomials attached to of order , which is denoted by , are defined as

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

where

In (2.13), we note that is symmetric in . From (2.13), we derive

It is easy to see that

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.

For , one has

Remark 2.2.

Let and in (1.4). Then we have

(see [25]).

We also calculate that

From the symmetric property of in and , we derive

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

Remark 2.4.

Let and in (2.23). We have

(see [25]).

## 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. - 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 Applications - 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. - 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) - 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.

## Acknowledgment

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

## Author information

### 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., Jang, L., Kim, Y. *et al.* On the Identities of Symmetry for the Generalized Bernoulli Polynomials Attached to of Higher Order.
*J Inequal Appl* **2009, **640152 (2009). https://doi.org/10.1155/2009/640152

Received:

Accepted:

Published:

### Keywords

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