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.
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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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ1_HTML.gif)
and that are called the
th Bernoulli numbers.
The recurrence formula for the classical Bernoulli numbers is as follows,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ2_HTML.gif)
(see [1, 3, 23]). The -extension of the following recurrence formula for the Bernoulli numbers is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ3_HTML.gif)
with the usual convention of replacing by
(see [5, 7, 14]).
Now, by introducing the following well-known identities
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ4_HTML.gif)
(see [6]).
The generating functions of the second kind Stirling numbers and -Bernstein polynomials, respectively, can be defined as follows,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ5_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ6_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ7_HTML.gif)
Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ8_HTML.gif)
where are classical Bernoulli polynomials. In the special case
,
are called the
th
-Bernoulli numbers. That is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ9_HTML.gif)
From (2.1) and (2.3), we note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ10_HTML.gif)
From (2.1) and (2.3), we can easily derive the following equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ11_HTML.gif)
Equations (2.4) and (2.5), we see that and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ12_HTML.gif)
Therefore, we obtain the following theorem.
Theorem 2.1.
For , one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ13_HTML.gif)
with the usual convention of replacing and
.
From (2.1), one notes that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ14_HTML.gif)
Therefore, one obtains the following theorem.
Theorem 2.2.
For , one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ15_HTML.gif)
By (2.1), one sees that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ16_HTML.gif)
By (2.1) and (2.10), one obtains the following theorem.
Theorem 2.3.
For , one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ17_HTML.gif)
From (2.11) one can derive that, for ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ18_HTML.gif)
By (2.12), one sees that, for ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ19_HTML.gif)
Therefore, one obtains the following theorem.
Theorem 2.4.
For , one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ20_HTML.gif)
In (2.9), substitute instead of
, one obtains
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ21_HTML.gif)
which is the relation between -Bernoulli polynomials,
-Bernoulli numbers, and
-Bernstein polynomials. In (1.5), substitute
instead of
, one gets
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ22_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ23_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ24_HTML.gif)
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ25_HTML.gif)
By (2.1) and (2.18), one sees that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ26_HTML.gif)
Therefore, one obtains the following theorem.
Theorem 2.6.
For and
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ27_HTML.gif)
By (2.18) and (2.19), one sees that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ28_HTML.gif)
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ29_HTML.gif)
For , one now considers the Mellin transformation for the generating function of
. That is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ30_HTML.gif)
for , and
.
From (2.24), one defines the zeta type function as follows,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ31_HTML.gif)
Note that is an analytic function in the whole complex
-plane. Using the Laurent series and the Cauchy residue theorem, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ32_HTML.gif)
By the same method, one can also obtain the following equations:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ33_HTML.gif)
For ,one defines Dirichlet type
-
-function as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ34_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ35_HTML.gif)
In (2.23), substitute instead of
, one obtains
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ36_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ37_HTML.gif)
One now defines particular -zeta function as follows,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ38_HTML.gif)
From (2.32), one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ39_HTML.gif)
where is given by (2.25). By (2.26), one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ40_HTML.gif)
Therefore, one obtains the following theorem.
Theorem 2.8.
For , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F951764/MediaObjects/13662_2010_Article_1354_Equ41_HTML.gif)
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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