Abstract

We give the twisted Carlitz's type -Bernoulli polynomials and numbers associated with -adic -inetgrals and discuss their properties. Furthermore, we define the multiple twisted Carlitz's type -Bernoulli polynomials and numbers and obtain the distribution relation for them.

1. Introduction

Let be a fixed odd prime number. Throughout this paper , , , and will, respectively, be the ring of -adic rational integers, the field of -adic rational numbers, the complex number field, and the -adic completion of the algebraic closure of . The -adic absolute value in is normalized so that . When one talks of -extension, is variously considered as an indeterminate, a complex number , or a -adic number . If , one normally assumes . If , one normally assumes that so that for each . We use the notation (cf. [120]) for all . For a fixed odd positive integer with , let where lies in . For any , is known to be a distribution on (cf. [120]).

We say that is uniformly differentiable function at a point and denote this property by , if the difference quotients have a limit as (cf. [1013]). The -adic -integral of a function was defined as

By using -adic -integrals on , it is well known that where . Then, we note that the Bernoulli numbers were defined as and hence, we have for all . For and , the multiple Bernoulli polynomials were defined as (cf. [2]). We note that From (1.9) and (1.10), we obtain In view of (1.11), the multiple Carlitz's type -Bernoulli polynomials were defined as In this case, , we write , which were called the Carlitz's type -Bernoulli numbers. By (1.11) and (1.12), we note that

In Section 2, we give the twisted Carlitz's type -Bernoulli polynomials and numbers associated with -adic -inetgrals and discuss their properties. In Section 3, we define the multiple twisted Carlitz's type -Bernoulli polynomials and numbers. We also obtain the distribution relation for them.

2. Twisted Carlitz's Type -Bernoulli Polynomials

In this section, we assume that with . By using -adic -integral on , we derive (cf. [8]), where . From (1.5), we can derive (cf. [8]), where and .

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

Now we define the twisted -Bernoulli polynomials as follows: We note that are called the twisted -Bernoulli numbers and by substituting , are the familiar Bernoulli numbers. By (2.3), we obtain the following Witt's type formula for the twisted -Bernoulli polynomials and numbers.

Theorem 2.1. For and , one has

From (2.5), we consider the twisted Carliz's type -Bernoulli polynomials by using -adic -integrals. For , we define the twisted Carlitz's type -Bernoulli polynomials as follows: When , we write which are called twisted Carlitz's type -Bernoulli numbers. Note that if , then . From (2.6), we can see that From (2.7), we can derive the generating function for the twisted Carlitz's type -Bernoulli polynomials as follows: Then it is easily to see that

By the th differentiation on both sides of (2.8) at , we also have for . We note that In view of (2.10), we define twisted Carlitz's type -zeta function as follows: for all and . We note that is analytic function in the whole complex -plane. We also have the following theorem in which twisted Carlitz's type -zeta functions interpolate twisted Carlitz's type -Bernoulli numbers and polynomials.

Theorem 2.2. For and , one has

From (2.11), we obtain the following distribution relation for the twisted -Bernoulli polynomials.

Theorem 2.3. For , , and , one has

Proof. If we put and and , then by (2.11), we have

3. Multiple Twisted Carlitz's Type -Bernoulli Polynomials

In this section, we consider the multiple twisted Carlitz's type -Bernoulli polynomials as follows: where , , and . We note that are called the multiple twisted Carlitz's type -Bernoulli numbers. We also obtain the generating function of the multiple twisted Carlitz's type -Bernoulli polynomials as follows: Finally, we have the following distribution relation for the multiple twisted -Bernoulli polynomials.

Theorem 3.1. For each , , and ,

Proof. If we put , and , then by (3.1), we have

Question 1. Are there the analytic multiple twisted Carlitz's type -zeta functions which interpolate multiple twisted Carlitz's type q-Bernoulli polynomials?