Open Access

A New Approach to -Bernoulli Numbers and -Bernoulli Polynomials Related to -Bernstein Polynomials

Advances in Difference Equations20112010:951764

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

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.

Many mathematicians have studied -Bernoulli, -Euler polynomials, and related topics (see [123]). It is known that the Bernoulli polynomials are defined by
(1.1)

and that are called the th Bernoulli numbers.

The recurrence formula for the classical Bernoulli numbers is as follows,
(1.2)
(see [1, 3, 23]). The -extension of the following recurrence formula for the Bernoulli numbers is
(1.3)

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

Now, by introducing the following well-known identities
(1.4)

(see [6]).

The generating functions of the second kind Stirling numbers and -Bernstein polynomials, respectively, can be defined as follows,
(1.5)
(1.6)

(see [2]), where (see [4]).

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 [719]).

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

Let be the set of natural numbers and . For with , let us define the -Bernoulli polynomials as follows,
(2.1)
Note that
(2.2)
where are classical Bernoulli polynomials. In the special case , are called the th -Bernoulli numbers. That is,
(2.3)
From (2.1) and (2.3), we note that
(2.4)
From (2.1) and (2.3), we can easily derive the following equation:
(2.5)
Equations (2.4) and (2.5), we see that and
(2.6)

Therefore, we obtain the following theorem.

Theorem 2.1.

For , one has
(2.7)

with the usual convention of replacing and .

From (2.1), one notes that
(2.8)

Therefore, one obtains the following theorem.

Theorem 2.2.

For , one has
(2.9)
By (2.1), one sees that
(2.10)

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

Theorem 2.3.

For , one has
(2.11)
From (2.11) one can derive that, for ,
(2.12)
By (2.12), one sees that, for ,
(2.13)

Therefore, one obtains the following theorem.

Theorem 2.4.

For , one has
(2.14)
In (2.9), substitute instead of , one obtains
(2.15)
which is the relation between -Bernoulli polynomials, -Bernoulli numbers, and -Bernstein polynomials. In (1.5), substitute instead of , one gets
(2.16)

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

Theorem 2.5.

For and , one has
(2.17)

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

Let be Dirichlet's character with . Then, one defines the generalized -Bernoulli polynomials attached to as follows,
(2.18)
In the special case , are called the th generalized -Bernoulli numbers attached to . Thus, the generating function of the generalized -Bernoulli numbers attached to are as follows,
(2.19)
By (2.1) and (2.18), one sees that
(2.20)

Therefore, one obtains the following theorem.

Theorem 2.6.

For and , one has
(2.21)
By (2.18) and (2.19), one sees that
(2.22)
Hence,
(2.23)
For , one now considers the Mellin transformation for the generating function of . That is,
(2.24)

for , and .

From (2.24), one defines the zeta type function as follows,
(2.25)
Note that is an analytic function in the whole complex -plane. Using the Laurent series and the Cauchy residue theorem, one has
(2.26)
By the same method, one can also obtain the following equations:
(2.27)
For ,one defines Dirichlet type - -function as
(2.28)
where . Note that is also a holomorphic function in the whole complex -plane. From the Laurent series and the Cauchy residue theorem, one can also derive the following equation:
(2.29)
In (2.23), substitute instead of , one obtains
(2.30)

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.

For and , one has
(2.31)
One now defines particular -zeta function as follows,
(2.32)
From (2.32), one has
(2.33)
where is given by (2.25). By (2.26), one has
(2.34)

Therefore, one obtains the following theorem.

Theorem 2.8.

For , we have
(2.35)

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science and Arts, University of Gaziantep

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© Mehmet Açikgöz et al. 2010

This article is published under license to BioMed Central Ltd. 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.