Multivariate Twisted -Adic -Integral on Associated with Twisted -Bernoulli Polynomials and Numbers

Abstract

Recently, many authors have studied twisted -Bernoulli polynomials by using the -adic invariant -integral on . In this paper, we define the twisted -adic -integral on and extend our result to the twisted -Bernoulli polynomials and numbers. Finally, we derive some various identities related to the twisted -Bernoulli polynomials.

1. Introduction

Let be a fixed prime number. Throughout this paper, the symbols and will denote the ring of rational integers, the ring of -adic integers, the field of -adic rational numbers, the complex number field, and the completion of the algebraic closure of respectively. Let be the set of natural numbers and Let be the normalized exponential valuation of with

When one talks of -extension, is variously considered as an indeterminate, a complex or -adic number If one normally assumes that If then we assume that .

For let be the -adic locally constant space defined by

(1.1)

where is the cyclic group of order

Let be the space of uniformly differentiable function on

For the -adic invariant -integral on is defined as

(1.2)

compare with [1–3].

It is well known that the twisted -Bernoulli polynomials of order are defined as

(1.3)

and are called the twisted -Bernoulli numbers of order When the polynomials and numbers are called the twisted -Bernoulli polynomials and numbers, respectively. When and the polynomials and numbers are called the twisted Bernoulli polynomials and numbers, respectively. When and the polynomials and numbers are called the ordinary Bernoulli polynomials and numbers, respectively.

Many authors have studied the twisted -Bernoulli polynomials by using the properties of the -adic invariant -integral on (cf. [4]). In this paper, we define the twisted -adic -integral on and extend our result to the twisted -Bernoulli polynomials and numbers. Finally, we derive some various identities related to the twisted -Bernoulli polynomials.

2. Multivariate Twisted -Adic -Integral on Associated withTwisted -Bernoulli Polynomials

In this section, we assume that with For , we define the -numbers as

(2.1)

Note that .

Let us define

(2.2)

where Note that

Now we construct the twisted -adic -integral on as follows:

(2.3)

where From the definition of the twisted -adic -integral on , we can consider the twisted -Bernoulli polynomials and numbers of order as follows:

(2.4)

In the special case in (2.4), are called the twisted -Bernoulli numbers of order .

If we take and in (2.4), we can easily see that

(2.5)

compare with [4].

Theorem 2.1.

For and we have

(2.6)

Moreover, if we take in Theorem 2.1, then we have the following identity for the twisted -Bernoull numbers

(2.7)

From the definition of multivariate twisted -adic -integral, we also see that

(2.8)

Corollary 2.2.

For and , one obtains

(2.9)

Note that

(2.10)

We have

(2.11)

It is easy to see that

(2.12)

By (2.11) and (2.12), we obtain the following theorem.

Theorem 2.3.

For and one has

(2.13)

Now we consider the modified extension of the twisted -Bernoulli polynomials of order as follows:

(2.14)

In the special case we write which are called the modified extension of the twisted -Bernoulli numbers of order

From (2.14), we derive that

(2.15)

Therefore, we obtain the following theorem.

Theorem 2.4.

For and one has

(2.16)

Now, we define as follows:

(2.17)

By (2.17), we can see that

(2.18)

Therefore, we obtain the following theorem.

Theorem 2.5.

For and one has

(2.19)

In (2.19), we can see the relations between the binomial coefficients and the modified extension of the twisted -Bernoulli polynomials of order