- Mehmet Açikgöz
^{1}Email author, - Dilek Erdal
^{1}and - Serkan Araci
^{1}

**2010**:951764

https://doi.org/10.1155/2010/951764

© Mehmet Açikgöz et al. 2010

**Received: **24 November 2010

**Accepted: **27 December 2010

**Published: **3 January 2011

## Abstract

We present a new generating function related to the -Bernoulli numbers and -Bernoulli polynomials. We give a new construction of these numbers and polynomials related to the second-kind Stirling numbers and -Bernstein polynomials. We also consider the generalized -Bernoulli polynomials attached to Dirichlet's character and have their generating function . We obtain distribution relations for the -Bernoulli polynomials and have some identities involving -Bernoulli numbers and polynomials related to the second kind Stirling numbers and -Bernstein polynomials. Finally, we derive the -extensions of zeta functions from the Mellin transformation of this generating function which interpolates the -Bernoulli polynomials at negative integers and is associated with -Bernstein polynomials.

## 1. Introduction, Definitions, and Notations

Let be the complex number field. We assume that with and that the -number is defined by in this paper.

and that are called the th Bernoulli numbers.

with the usual convention of replacing by (see [5, 7, 14]).

(see [6]).

Throughout this paper, , , , , and will respectively denote the ring of rational integers, the field of rational numbers, the ring -adic rational integers, the field of -adic rational numbers, and the completion of the algebraic closure of . Let be the normalized exponential valuation of such that . If , we normally assume or so that for (see [7–19]).

In this study, we present a new generating function related to the -Bernoulli numbers and -Bernoulli polynomials and give a new construction of these numbers and polynomials related to the second kind Stirling numbers and -Bernstein polynomials. We also consider the generalized -Bernoulli polynomials attached to Dirichlet's character and have their generating function. We obtain distribution relations for the -Bernoulli polynomials and have some identities involving -Bernoulli numbers and polynomials related to the second kind Stirling numbers and -Bernstein polynomials. Finally, we derive the -extensions of zeta functions from the Mellin transformation of this generating function which interpolates the -Bernoulli polynomials at negative integers and are associated with -Bernstein polynomials.

## 2. New Approach to -Bernoulli Numbers and Polynomials

Therefore, we obtain the following theorem.

Theorem 2.1.

with the usual convention of replacing and .

Therefore, one obtains the following theorem.

Theorem 2.2.

By (2.1) and (2.10), one obtains the following theorem.

Theorem 2.3.

Therefore, one obtains the following theorem.

Theorem 2.4.

In (2.16), substitute instead of , and putting the result in (2.15), one has the following theorem.

Theorem 2.5.

where and are the second kind Stirling numbers and -Bernstein polynomials, respectively.

Therefore, one obtains the following theorem.

Theorem 2.6.

which is the relation between the th generalized -Bernoulli numbers and -Bernoulli polynomials attached to and -Bernstein polynomials. From (2.16), one has the following theorem.

Theorem 2.7.

Therefore, one obtains the following theorem.

Theorem 2.8.

## Authors’ Affiliations

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