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

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]).

For , we set

(1.1)

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

(1.2)

(1.3)

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

(1.4)

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

(1.5)

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

(1.6)

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

(2.1)

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

(2.2)

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

(2.3)

(2.4)

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

(2.5)

By (1.5), we can see that

(2.6)

In (1.4), it is easy to show that

(2.7)

For each integer , let

(2.8)

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

(2.9)

From (2.9), we note that

(2.10)

for all . Let ; then we have

(2.11)

By (2.9), we see that

(2.12)

Let be as follows:

(2.13)

Then we have

(2.14)

From (2.13), we derive

(2.15)

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

(2.16)

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

(2.17)

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 ,

(2.18)

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

(2.19)

where is the binomial coefficient.

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

(2.20)

From the symmetry of , we can also derive

(2.21)

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

(2.22)

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

(2.23)

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

(3.1)

(3.2)

(cf. [2, 18, 23, 27]).

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

(3.3)

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

(3.4)

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

(3.5)

(3.6)

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

(3.7)

Thus we have

(3.8)

Then

(3.9)

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

(3.10)

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

(3.11)

Thus,

(3.12)

for all . This means that

(3.13)

for all . For all , we have

(3.14)

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

(3.15)

(3.16)

for each . For , we set

(3.17)

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

(3.18)

Thus we can obtain

(3.19)

From (3.19), we derive

(3.20)

By the symmetry of in and , we can see that

(3.21)

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

Theorem 3.1.

For , , one has

(3.22)

Remark 3.2.

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

(3.23)

Now we can also calculate

(3.24)

From the symmetric property of in and , we derive

(3.25)

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

Theorem 3.3.

For , , we have

(3.26)

Remark 3.4.

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

(3.27)

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.