# Some Identities on the Generalized -Bernoulli Numbers and Polynomials Associated with -Volkenborn Integrals

- T Kim
^{1}Email author, - J Choi
^{1}, - B Lee
^{2}and - CS Ryoo
^{3}

**2010**:575240

https://doi.org/10.1155/2010/575240

© T. Kim et al. 2010

**Received: **23 August 2010

**Accepted: **30 September 2010

**Published: **11 October 2010

## Abstract

## Keywords

## 1. Introduction

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

with the usual convention of replacing by , (see [1–13]).

(see [1]), where are called the th Carlitz's -Bernoulli polynomials (see [1, 12, 13]).

(see [13]). Recently, many authors have studied in the different several areas related to -theory (see [1–13]). In this paper, we present a systemic study of some families of multiple Carlitz's type -Bernoulli numbers and polynomials by using the integral equations of -adic -integrals on . First, we derive some interesting equations of -adic -integrals on . From these equations, we give some interesting formulae for the higher-order Carlitz's type -Bernoulli numbers and polynomials in the -adic number field.

## 2. On the Generalized Higher-Order -Bernoulli Numbersand Polynomials

where are called the th ordinary Bernoulli polynomials. In the special case, , are called the th -Bernoulli numbers.

By (2.2), we have the following lemma.

Lemma 2.1.

By (2.5) and (2.7), one obtains the following theorem.

Theorem 2.2.

In the special case, , the sequence are called the th generalized -Bernoulli numbers attached to .

where are called the th generalized -Bernoulli polynomials of order attaches to .

In the special case, , the sequence are called the th generalized -Bernoulli numbers of order attaches to .

By (2.13) and (2.14), one obtains the following theorem.

Theorem 2.3.

In the special case, , are called the th -Bernoulli numbers of order .

By (2.17), one obtains the following theorem.

Theorem 2.4.

In the special case, , are called the th generalized -Bernoulli numbers attached to of order .

By (2.23), (2.25), and (2.26), one obtains the following theorem.

Theorem 2.5.

By (2.29), one obtains the following theorem.

Theorem 2.6.

Therefore, one obtains the following corollary.

Corollary 2.7.

By (2.35) and (2.36), one obtains the following corollary.

Corollary 2.8.

Thus, one obtains the following theorem.

Theorem 2.9.

By (2.46), (2.48), and (2.50), one obtains the following theorem.

Theorem 2.10.

where are the th ordinary Bernoulli polynomials.

Therefore, one obtains the following theorem.

Theorem 2.11.

## Declarations

### Acknowledgments

The authors express their gratitude to The referees for their valuable suggestions and comments. This paper was supported by the research grant of Kwangwoon University in 2010.

## Authors’ Affiliations

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