# A Note on Symmetric Properties of the Twisted -Bernoulli Polynomials and the Twisted Generalized -Bernoulli Polynomials

- L.-C. Jang
^{1}, - H. Yi
^{2}Email author, - K. Shivashankara
^{3}, - T. Kim
^{4}, - Y. H. Kim
^{4}and - B. Lee
^{5}

**2010**:801580

**DOI: **10.1155/2010/801580

© L.-C. Jang et al. 2010

**Received: **11 September 2009

**Accepted: **31 May 2010

**Published: **24 June 2010

## Abstract

We define the twisted -Bernoulli polynomials and the twisted generalized -Bernoulli polynomials attached to of higher order and investigate some symmetric properties of them. Furthermore, using these symmetric properties of them, we can obtain some relationships between twisted -Bernoulli numbers and polynomials and between twisted generalized -Bernoulli numbers and polynomials.

## 1. Introduction

Let be a fixed prime number. Throughout this paper , and will, respectively, denote the ring of -adic rational integers, the field of -adic rational numbers, and the completion of algebraic closure of . Let be the normalized exponential valuation of with When one talks of -extension, is variously considered as an indeterminate, a complex number or a -adic number . If one normally assumes If then we assume so that for (cf. [1–32]).

In this paper, we define the twisted -Bernoulli polynomials and the twisted generalized -Bernoulli polynomials attached to of higher order and investigate some symmetric properties of them. Furthermore, using these symmetric properties of them, we can obtain some relationships between the twisted -Bernoulli numbers and polynomials and between the twisted generalized -Bernoulli numbers and polynomials attached to of higher order.

## 2. The Twisted -Bernoulli Polynomials

By comparing the coefficients of on both sides of (2.16) and (2.17), we obtain the following theorem.

Theorem 2.1.

where is the binomial coefficient.

From Theorem 2.1, if we take , then we have the following corollary.

Corollary 2.2.

where is the binomial coefficient.

By comparing the coefficients of on both sides of (2.20) and (2.21), we obtain the following theorem.

Theorem 2.3.

We note that by setting in Theorem 2.3, we get the following multiplication theorem for the twisted -Bernoulli polynomials.

Theorem 2.4.

Remark 2.5.

[18], Kim suggested open questions related to finding symmetric properties for Carlitz -Bernoulli numbers. In this paper, we give the symmetric property for -Bernoulli numbers in the viewpoint to give the answer of Kim's open questions.

## 3. The Twisted Generalized Bernoulli Polynomials Attached to of Higher Order

By comparing the coefficients on both sides of (3.20) and (3.21), we see the following theorem.

Theorem 3.1.

Remark 3.2.

By comparing the coefficients on both sides of (3.24) and (3.26), we obtain the following theorem.

Theorem 3.3.

Remark 3.4.

Remark 3.5.

In our results for , we can also derive similar results, which were treated in [27]. In this paper, we used the -adic integrals to derive the symmetric properties of the -Bernoulli polynomials. By using the symmetric properties of -adic integral on , we can easily derive many interesting symmetric properties related to Bernoulli numbers and polynomials.

## Declarations

### Acknowledgments

The authors express Their sincere gratitude to referees for their valuable suggestions and comments. This work has been conducted by the Research Grant of Kwangwoon University in 2010.

## Authors’ Affiliations

## References

- 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 - Jang L-C:
**On a****-analogue of the****-adic generalized twisted****-functions and**-adic**-integrals.***Journal of the Korean Mathematical Society*2007,**44**(1):1-10. 10.4134/JKMS.2007.44.1.139MathSciNetView ArticleMATHGoogle Scholar - Jang L-C:
**Multiple twisted****-Euler numbers and polynomials associated with**-adic**-integrals.***Advances in Difference Equations*2008,**2008:**-11.Google 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 - Jang L-C, Kim T:
**On the distribution of the****-Euler polynomials and the****-Genocchi polynomials of higher order.***Journal of Inequalities and Applications*2008,**2008:**-9.Google Scholar - Kim T:
**-Volkenborn integration.***Russian Journal of Mathematical Physics*2002,**9**(3):288-299.MathSciNetMATHGoogle Scholar - Kim T:
**On Euler-Barnes multiple zeta functions.***Russian Journal of Mathematical Physics*2003,**10**(3):261-267.MathSciNetMATHGoogle Scholar - Kim T:
**Analytic continuation of multiple****-zeta functions and their values at negative integers.***Russian Journal of Mathematical Physics*2004,**11**(1):71-76.MathSciNetMATHGoogle Scholar - Kim T:
**Power series and asymptotic series associated with the****-analog of the two-variable****-adic****-function.***Russian Journal of Mathematical Physics*2005,**12**(2):186-196.MathSciNetMATHGoogle Scholar - Kim T:
**Multiple**-adic**-function.***Russian Journal of Mathematical Physics*2006,**13**(2):151-157. 10.1134/S1061920806020038MathSciNetView ArticleMATHGoogle Scholar - Kim T:
**A note on**-Adic**-integral on Zp associated with****-Euler numbers.***Advanced Studies in Contemporary Mathematics*2007,**15:**133-138.MATHGoogle Scholar - Kim T:
**On****-adic interpolating function for****-Euler numbers and its derivatives.***Journal of Mathematical Analysis and Applications*2008,**339**(1):598-608. 10.1016/j.jmaa.2007.07.027MathSciNetView ArticleMATHGoogle Scholar - Kim T:
**On the analogs of Euler numbers and polynomials associated with**-adic -integral on at**.***Journal of Mathematical Analysis and Applications*2007,**331**(2):779-792. 10.1016/j.jmaa.2006.09.027MathSciNetView 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:
-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 some formulae for the****-Euler numbers and polynomials.***Proc. Jangjeon Math. Soc.*2006,**9**(2):227-232.MathSciNetMATHGoogle 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:
**On the symmetries of the****-Bernoulli polynomials.***Abstract and Applied Analysis*2008,**2008:**-7.Google Scholar - Kim T:
**Note on Dedekind type DC sums.***Advanced Studies in Contemporary Mathematics*2009,**18**(2):249-260.MathSciNetMATHGoogle Scholar - Kim T, Jang L-C, Pak HK:
**A note on****-Euler and Genocchi numbers.***Proceedings of the Japan Academy, Series A*2001,**77**(8):139-141. 10.3792/pjaa.77.139MathSciNetView ArticleMATHGoogle Scholar - Kim T:
**Note on the****-Euler numbers of higher order.***Advanced Studies in Contemporary Mathematics*2009,**19**(1):25-29.MathSciNetMATHGoogle Scholar - Kim T, Kim M-S, Jang L-C, Rim S-H:
**New****-Euler numbers and polynomials associated with**-adic**-integrals.***Advanced Studies in Contemporary Mathematics*2007,**15**(2):243-252.MathSciNetMATHGoogle Scholar - Kim W, Kim Y-H, Jang L-C:
**On the****-extension of apostol-euler numbers and polynomials.***Abstract and Applied Analysis*2008,**2008:**-10.Google 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 - Ozden H, Simsek Y, Rim S-H, Cangul IN:
**A note on**-adic**-Euler measure.***Advanced Studies in Contemporary Mathematics*2007,**14:**233-239.MathSciNetGoogle Scholar - Rim S-H, Kim Y-H, Lee BJ, Kim T:
**Some identities of the generalized twisted Bernoulli numbers and polynomials of higher order.***Journal of Computational Analysis and Applications*2010,**12:**695-702.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:
**Note on the Euler****-zeta functions.***Journal of Number Theory*2009,**129**(7):1798-1804. 10.1016/j.jnt.2008.10.007MathSciNetView ArticleMATHGoogle Scholar - Kim T:
**A new approach to**-adic -**-functions.***Advanced Studies in Contemporary Mathematics*2006,**12**(1):61-72.MathSciNetMATHGoogle Scholar - Kim T, Rim S-H:
**On the twisted****-Euler numbers and polynomials associated with basic**-**-functions.***Journal of Mathematical Analysis and Applications*2007,**336**(1):738-744. 10.1016/j.jmaa.2007.03.035MathSciNetView ArticleMATHGoogle Scholar - Kim T:
**New approach to****-Euler polynomials of higher order.***Russian Journal of Mathematical Physics*2010,**17**(2):201-207. 10.1134/S1061920810020068View ArticleGoogle Scholar - Kim T:
**Barnes-type multiple****-zeta functions and****-Euler polynomials.***Journal of Physics A*2010,**43:**-11.Google Scholar

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