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

*Advances in Difference Equations*
**volumeÂ 2010**, ArticleÂ number:Â 801580 (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. [1â€“32]).

For , we set

(see [1â€“13]). The Bernoulli numbers and polynomials are defined by the generating function as

(cf. [17, 18, 21, 24, 26]). Let be the set of uniformly differentiable functions on . For , the -adic invariant integral on is defined as

Note that (see [27]). Let be a translation with . We note that

(cf. [1â€“32]). Kim [18] studied the symmetric properties of the -Bernoulli numbers and polynomials as follows:

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

Let be the locally constant space, where is the cyclic group of order . For , we denote the locally constant function by

(cf. [2, 3, 21, 24]). If we take , then

Now we define the -extension of twisted Bernoulli numbers and polynomials as follows:

(see [31]). From (1.5), (2.2), (2.3), and (2.4), we can derive

By (1.5), we can see that

In (1.4), it is easy to show that

For each integer , let

From (2.6), (2.7), and (2.8), we derive

From (2.9), we note that

for all . Let ; then we have

By (2.9), we see that

Let be as follows:

Then we have

From (2.13), we derive

By (2.4), (2.12), and (2.15), we can see that

By the symmetry of -adic invariant integral on , we also see that

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

Theorem 2.1.

Let . Then for all ,

where is the binomial coefficient.

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

Corollary 2.2.

For , one we has

where is the binomial coefficient.

By (2.17), (2.18), and (2.19), we can see that

From the symmetry of , we can also derive

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

Theorem 2.3.

For , , we have

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

Theorem 2.4.

For , , one has

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

In this section, we consider the generalized Bernoulli numbers and polynomials and then define the twisted generalized Bernoulli polynomials attached to of higher order by using multivariate -adic invariant integrals on . Let be Dirichlet's character with conductor . Then the generalized Bernoulli numbers and polynomials attached to are defined as

Let be the locally constant space, where is the cyclic group of order . For , we denote the locally constant function by

(cf. [2, 3, 21, 23, 24]). If we take , for with , then it is obvious from (3.1) that

Now we define the twisted generalized Bernoulli numbers and polynomials attached to as follows:

for each (see [31, 32]). By (3.5) and (3.6),

Thus we have

Then

Let us define the -adic twisted -function as follows:

By (3.9) and (3.10), we see that

Thus,

for all . This means that

for all . For all , we have

The twisted generalized Bernoulli numbers and polynomials attached to of order are defined as

for each . For , we set

where . In (3.17), we note that is symmetric in . From (3.17), we have

Thus we can obtain

From (3.19), we derive

By the symmetry of in and , we can see that

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

Theorem 3.1.

For , , one has

Remark 3.2.

If we take and in (3.22), then we have

Now we can also calculate

From the symmetric property of in and , we derive

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

Theorem 3.3.

For , , we have

Remark 3.4.

If we take and in (3.26), then one has

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.

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## 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.

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Jang, LC., Yi, H., Shivashankara, K. *et al.* A Note on Symmetric Properties of the Twisted -Bernoulli Polynomials and the Twisted Generalized -Bernoulli Polynomials.
*Adv Differ Equ* **2010**, 801580 (2010). https://doi.org/10.1155/2010/801580

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DOI: https://doi.org/10.1155/2010/801580

### Keywords

- Ordinary Differential Equation
- Functional Equation
- Prime Number
- Rational Number
- Constant Function