# On the Symmetric Properties for the Generalized Twisted Bernoulli Polynomials

- Taekyun Kim
^{1}and - Young-Hee Kim
^{1}Email author

**2009**:164743

https://doi.org/10.1155/2009/164743

© T. Kim and Y.-H. Kim. 2009

**Received: **6 July 2009

**Accepted: **18 October 2009

**Published: **19 October 2009

## Abstract

## Keywords

## 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

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

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

Let be a fixed positive integer. For , let

where lies in . It is easy to see that

The ordinary Bernoulli polynomials are defined as

and the Bernoulli numbers are defined as (see [1–19]).

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

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

and the twisted Bernoulli numbers are defined as (see [15–18]).

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

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, 20–27]). 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. [15–18]). 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 [11–13]).

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

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

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

From (2.3), we derive that

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

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

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

Theorem 2.1.

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

where . Taking in (2.8), it follows that

Thus, we have

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

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

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

Theorem 2.2.

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

Now let us define the -adic functional as follows:

Then it follows from (2.14) that

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

On the other hand, the symmetric property of shows that

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

Theorem 2.3.

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

Corollary 2.4.

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

From the symmetric property of , we also see that

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

Theorem 2.5.

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

## Declarations

### Acknowledgment

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

## Authors’ Affiliations

## References

- Kim T:
**-Volkenborn integration.***Russian Journal of Mathematical Physics*2002,**9**(3):288–299.MathSciNetMATHGoogle Scholar - Kim T:
**On the symmetries of the****-Bernoulli polynomials.***Abstract and Applied Analysis*2008,**2008:**-7.Google Scholar - 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 - 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:
**Note on****-Genocchi numbers and polynomials.***Advanced Studies in Contemporary Mathematics*2008,**17**(1):9–15.MathSciNetMATHGoogle Scholar - 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 - 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 - Carlitz L:
**-Bernoulli numbers and polynomials.***Duke Mathematical Journal*1948,**15:**987–1000. 10.1215/S0012-7094-48-01588-9MathSciNetView ArticleMATHGoogle Scholar - 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 - 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 - 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 - 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 - 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 - 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 - Simsek Y:
**Theorems on twisted****-function and twisted Bernoulli numbers.***Advanced Studies in Contemporary Mathematics*2005,**11**(2):205–218.MathSciNetMATHGoogle Scholar - 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:
**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 - 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 - 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 - 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 - 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 - Kim T:
**Note on Dedekind type DC sums.***Advanced Studies in Contemporary Mathematics*2009,**18**(2):249–260.MathSciNetMATHGoogle Scholar - Kim T:
**A note on the generalized****-Euler numbers.***Proceedings of the Jangjeon Mathematical Society*2009,**12**(1):45–50.MathSciNetMATHGoogle 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:
**Symmetry identities for the twisted generalized Euler polynomials.***Advanced Studies in Contemporary Mathematics*2009,**19**(2):151–155.MathSciNetMATHGoogle Scholar - 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 - 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

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.