#### Abstract

We discuss a new concept of the -extension of Bernoulli measure. From those measures, we derive some interesting properties on the generalized -Bernoulli numbers with weight attached to .

#### 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 set of natural numbers and . Let be the normalized exponential valuation of with (see [1–14]).

When we talk of -extension, is variously considered as an indeterminate, a complex number , or a -adic number . Throughout this paper we assume that with , and we use the notation of -number as (see [1–14]). Thus, we note that .

In [2], Carlitz defined a set of numbers inductively by with the usual convention of replacing by .

These numbers are -extension of ordinary Bernoulli numbers . But they do not remain finite when . So he modified (1.2) as follows: with the usual convention of replacing by .

The numbers are called the -th Carlitz -Bernoulli numbers.

In [1], Carlitz also considered the extended Carlitz’s -Bernoulli numbers as follows: with the usual convention of replacing by .

Recently, Kim considered -Bernoulli numbers, which are different extended Carlitz’s -Bernoulli numbers, as follows: for and , with the usual convention of replacing by (see [3]).

The numbers are called the -th -Bernoulli numbers with weight .

For fixed with , we set where satisfies the condition .

Let be the space of uniformly differentiable functions on . For , the -adic -integral on is defined by Kim as follows: (see [3, 4, 15, 16]). By (1.5) and (1.7), the Witt’s formula for the -Bernoulli numbers with weight is given by

The -Bernoulli polynomials with weight are also defined by

From (1.7), (1.8), and (1.9), we can derive the Witt’s formula for as follows:

For and , the distribution relation for the -Bernoulli polynomials with weight are known that (see [3]). Recently, several authors have studied the -adic -Euler and Bernoulli measures on (see [8, 9, 11, 13, 14]). The purpose of this paper is to construct -adic -Bernoulli distribution with weight (-adic -Bernoulli unbounded measure with weight ) on and to study their integral representations. Finally, we construct the generalized -Bernoulli numbers with weight and investigate their properties related to -adic --functions.

#### 2. -Adic -Bernoulli Distribution with Weight

Let be any compact-open subset of , such as or . A -adic distribution on is defined to be an additive map from the collection of compact open set in to : where is any collection of disjoint compact opensets in .

The set has a topological basis of compact open sets of the form .

Consequently, if is any compact open subset of , it can be written as a finite disjoint union of sets where and with for .

Indeed, the -adic ball can be represented as the union of smaller balls

Lemma 2.1. *Every map from the collection of compact-open sets in to for which
**
holds whenever , extends to a -adic distribution (-adic unbounded measure) on .*

Now we define a map on the balls in as follows: where is the unique number in the set such that .

If , then

From (2.6), we note that is -adic distribution on if and only if

Theorem 2.2. *Let and . Then we see that is -adic distribution on if and only if
**One sets
**From (2.5) and (2.9), one gets
*

By (1.11), (2.10), and Theorem 2.2, we obtain the following theorem.

Theorem 2.3. *Let be given by
**
Then extends to a -valued distribution on the compact open sets .**From (2.11), one notes that
**By (1.11) and (2.12), one gets
*

Therefore, we obtain the following theorem.

Theorem 2.4. *For and , one has
**Let be Dirichlet character with conductor . Then one defines the generalized -Bernoulli numbers attached to as follows:
**From (2.11) and (2.15), one can derive the following equation;
**For , one has
*

Therefore, we obtain the following theorem.

Theorem 2.5. *For , one has
**
Define
**
By a simple calculation, one gets
**
By (2.19) and (2.20), one gets
**Now one defines the operator on by
**Thus, by (2.22), one gets
**
Let us define . Then one has
**
From the definition of , one can easily derive the following equation;
**
Let . Then one gets
**By (2.21) and (2.26), one obtains the following equation:
**
where .*