#### 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 ).

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 ). Thus, we note that .

In , 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 , 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 ).

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 ). 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 .