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A New Approach to -Bernoulli Numbers and -Bernoulli Polynomials Related to -Bernstein Polynomials
Advances in Difference Equations volume 2010, Article number: 951764 (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 [1–23]). It is known that the Bernoulli polynomials are defined by
and that are called the th Bernoulli numbers.
The recurrence formula for the classical Bernoulli numbers is as follows,
(see [1, 3, 23]). The -extension of the following recurrence formula for the Bernoulli numbers is
with the usual convention of replacing by (see [5, 7, 14]).
Now, by introducing the following well-known identities
(see [6]).
The generating functions of the second kind Stirling numbers and -Bernstein polynomials, respectively, can be defined as follows,
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
Let be the set of natural numbers and . For with , let us define the -Bernoulli polynomials as follows,
Note that
where are classical Bernoulli polynomials. In the special case , are called the th -Bernoulli numbers. That is,
From (2.1) and (2.3), we note that
From (2.1) and (2.3), we can easily derive the following equation:
Equations (2.4) and (2.5), we see that and
Therefore, we obtain the following theorem.
Theorem 2.1.
For , one has
with the usual convention of replacing and .
From (2.1), one notes that
Therefore, one obtains the following theorem.
Theorem 2.2.
For , one has
By (2.1), one sees that
By (2.1) and (2.10), one obtains the following theorem.
Theorem 2.3.
For , one has
From (2.11) one can derive that, for ,
By (2.12), one sees that, for ,
Therefore, one obtains the following theorem.
Theorem 2.4.
For , one has
In (2.9), substitute instead of , one obtains
which is the relation between -Bernoulli polynomials, -Bernoulli numbers, and -Bernstein polynomials. In (1.5), substitute instead of , one gets
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
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,
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,
By (2.1) and (2.18), one sees that
Therefore, one obtains the following theorem.
Theorem 2.6.
For and , one has
By (2.18) and (2.19), one sees that
Hence,
For , one now considers the Mellin transformation for the generating function of . That is,
for , and .
From (2.24), one defines the zeta type function as follows,
Note that is an analytic function in the whole complex -plane. Using the Laurent series and the Cauchy residue theorem, one has
By the same method, one can also obtain the following equations:
For ,one defines Dirichlet type --function as
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:
In (2.23), substitute instead of , one obtains
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
One now defines particular -zeta function as follows,
From (2.32), one has
where is given by (2.25). By (2.26), one has
Therefore, one obtains the following theorem.
Theorem 2.8.
For , we have
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Açikgöz, M., Erdal, D. & Araci, S. A New Approach to -Bernoulli Numbers and -Bernoulli Polynomials Related to -Bernstein Polynomials. Adv Differ Equ 2010, 951764 (2011). https://doi.org/10.1155/2010/951764
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DOI: https://doi.org/10.1155/2010/951764