Abstract and Applied Analysis

Volume 2011 (2011), Article ID 867217, 9 pages

http://dx.doi.org/10.1155/2011/867217

## A Note on the Generalized -Bernoulli Measures with Weight

^{1}Division of General Education, Kwangwoon University, Seoul 139-701, Republic of Korea^{2}Hanrimwon, Kwangwoon University, Seoul 139-701, Republic of Korea^{3}Department of Mathematics, Hannam University, Daejeon 306-791, Republic of Korea

Received 21 April 2011; Accepted 16 May 2011

Academic Editor: Gabriel Turinici

Copyright © 2011 T. Kim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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 .*

#### References

- L. Carlitz, “Expansions of $q$-Bernoulli numbers,”
*Duke Mathematical Journal*, vol. 25, pp. 355–364, 1958. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L. Carlitz, “$q$-Bernoulli numbers and polynomials,”
*Duke Mathematical Journal*, vol. 15, pp. 987–1000, 1948. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Kim, “On the weighted $q$-Bernoulli numbers and polynomials,”
*Advanced Studies in Contemporary Mathematics*, vol. 21, pp. 207–215, 2011. View at Google Scholar - T. Kim, “On a $q$-analogue of the $p$-adic log gamma functions and related integrals,”
*Journal of Number Theory*, vol. 76, no. 2, pp. 320–329, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y.-H. Kim, “On the $p$-adic interpolation functions of the generalized twisted $(h,q)$-Euler numbers,”
*International Journal of Mathematical Analysis*, vol. 3, no. 17–20, pp. 897–904, 2009. View at Google Scholar - A. Bayad, “Modular properties of elliptic Bernoulli and Euler functions,”
*Advanced Studies in Contemporary Mathematics*, vol. 20, no. 3, pp. 389–401, 2010. View at Google Scholar - A. Bayad, “Arithmetical properties of elliptic Bernoulli and Euler numbers,”
*International Journal of Algebra*, vol. 4, no. 5-8, pp. 353–372, 2010. View at Google Scholar - H. Ozden, Y. Simsek, S.-H. Rim, and I. N. Cangul, “A note on $p$-adic $q$-Euler measure,”
*Advanced Studies in Contemporary Mathematics*, vol. 14, no. 2, pp. 233–239, 2007. View at Google Scholar - H. Ozden, Y. Simsek, and I. N. Cangul, “Euler polynomials associated with $p$-adic $q$-Euler measure,”
*General Mathematics*, vol. 15, no. 2, pp. 24–37, 2007. View at Google Scholar - Y. Simsek, “Special functions related to Dedekind-type DC-sums and their applications,”
*Russian Journal of Mathematical Physics*, vol. 17, no. 4, pp. 495–508, 2010. View at Publisher · View at Google Scholar - S.-H. Rim and T. Kim, “A note on $p$-adic Euler measure on ${\mathbb{Z}}_{p}$,”
*Russian Journal of Mathematical Physics*, vol. 13, no. 3, pp. 358–361, 2006. View at Publisher · View at Google Scholar · View at MathSciNet - S.-H. Rim, J.-H. Jin, E.-J. Moon, and S.-J. Lee, “Some identities on the $q$-Genocchi polynomials of higher-order and $q$-Stirling numbers by the fermionic $p$-adic integral on ${\mathbb{Z}}_{p}$,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 2010, Article ID 860280, 14 pages, 2010. View at Publisher · View at Google Scholar - N. Koblitz, “On Carlitz's $q$-Bernoulli numbers,”
*Journal of Number Theory*, vol. 14, no. 3, pp. 332–339, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Kim, “On explicit formulas of $p$-adic $q$-$L$-functions,”
*Kyushu Journal of Mathematics*, vol. 48, no. 1, pp. 73–86, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Kim, “$q$-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients,”
*Russian Journal of Mathematical Physics*, vol. 15, no. 1, pp. 51–57, 2008. View at Google Scholar · View at Zentralblatt MATH - T. Kim, “$q$-Volkenborn integration,”
*Russian Journal of Mathematical Physics*, vol. 9, no. 3, pp. 288–299, 2002. View at Google Scholar · View at Zentralblatt MATH