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Multivariate Twisted
-Adic
-Integral on
Associated with Twisted
-Bernoulli Polynomials and Numbers
Journal of Inequalities and Applications volume 2010, Article number: 579509 (2010)
Abstract
Recently, many authors have studied twisted -Bernoulli polynomials by using the
-adic invariant
-integral on
. In this paper, we define the twisted
-adic
-integral on
and extend our result to the twisted
-Bernoulli polynomials and numbers. Finally, we derive some various identities related to the twisted
-Bernoulli polynomials.
1. Introduction
Let be a fixed prime number. Throughout this paper, the symbols
and
will denote the ring of rational integers, the ring of
-adic integers, the field of
-adic rational numbers, the complex number field, and the completion of the algebraic closure of
respectively. Let
be the set of natural numbers and
Let
be the normalized exponential valuation of
with
When one talks of -extension,
is variously considered as an indeterminate, a complex
or
-adic number
If
one normally assumes that
If
then we assume that
.
For let
be the
-adic locally constant space defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F579509/MediaObjects/13660_2010_Article_2191_Equ1_HTML.gif)
where is the cyclic group of order
Let be the space of uniformly differentiable function on
For the
-adic invariant
-integral on
is defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F579509/MediaObjects/13660_2010_Article_2191_Equ2_HTML.gif)
It is well known that the twisted -Bernoulli polynomials of order
are defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F579509/MediaObjects/13660_2010_Article_2191_Equ3_HTML.gif)
and are called the twisted
-Bernoulli numbers of order
When
the polynomials and numbers are called the twisted
-Bernoulli polynomials and numbers, respectively. When
and
the polynomials and numbers are called the twisted Bernoulli polynomials and numbers, respectively. When
and
the polynomials and numbers are called the ordinary Bernoulli polynomials and numbers, respectively.
Many authors have studied the twisted -Bernoulli polynomials by using the properties of the
-adic invariant
-integral on
(cf. [4]). In this paper, we define the twisted
-adic
-integral on
and extend our result to the twisted
-Bernoulli polynomials and numbers. Finally, we derive some various identities related to the twisted
-Bernoulli polynomials.
2. Multivariate Twisted
-Adic
-Integral on
Associated withTwisted
-Bernoulli Polynomials
In this section, we assume that with
For
, we define the
-numbers as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F579509/MediaObjects/13660_2010_Article_2191_Equ4_HTML.gif)
Note that .
Let us define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F579509/MediaObjects/13660_2010_Article_2191_Equ5_HTML.gif)
where Note that
Now we construct the twisted -adic
-integral on
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F579509/MediaObjects/13660_2010_Article_2191_Equ6_HTML.gif)
where From the definition of the twisted
-adic
-integral on
, we can consider the twisted
-Bernoulli polynomials and numbers of order
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F579509/MediaObjects/13660_2010_Article_2191_Equ7_HTML.gif)
In the special case in (2.4),
are called the twisted
-Bernoulli numbers of order
.
If we take and
in (2.4), we can easily see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F579509/MediaObjects/13660_2010_Article_2191_Equ8_HTML.gif)
compare with [4].
Theorem 2.1.
For and
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F579509/MediaObjects/13660_2010_Article_2191_Equ9_HTML.gif)
Moreover, if we take in Theorem 2.1, then we have the following identity for the twisted
-Bernoull numbers
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F579509/MediaObjects/13660_2010_Article_2191_Equ10_HTML.gif)
From the definition of multivariate twisted -adic
-integral, we also see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F579509/MediaObjects/13660_2010_Article_2191_Equ11_HTML.gif)
Corollary 2.2.
For and
, one obtains
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F579509/MediaObjects/13660_2010_Article_2191_Equ12_HTML.gif)
Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F579509/MediaObjects/13660_2010_Article_2191_Equ13_HTML.gif)
We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F579509/MediaObjects/13660_2010_Article_2191_Equ14_HTML.gif)
It is easy to see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F579509/MediaObjects/13660_2010_Article_2191_Equ15_HTML.gif)
By (2.11) and (2.12), we obtain the following theorem.
Theorem 2.3.
For and
one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F579509/MediaObjects/13660_2010_Article_2191_Equ16_HTML.gif)
Now we consider the modified extension of the twisted -Bernoulli polynomials of order
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F579509/MediaObjects/13660_2010_Article_2191_Equ17_HTML.gif)
In the special case we write
which are called the modified extension of the twisted
-Bernoulli numbers of order
From (2.14), we derive that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F579509/MediaObjects/13660_2010_Article_2191_Equ18_HTML.gif)
Therefore, we obtain the following theorem.
Theorem 2.4.
For and
one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F579509/MediaObjects/13660_2010_Article_2191_Equ19_HTML.gif)
Now, we define as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F579509/MediaObjects/13660_2010_Article_2191_Equ20_HTML.gif)
By (2.17), we can see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F579509/MediaObjects/13660_2010_Article_2191_Equ21_HTML.gif)
Therefore, we obtain the following theorem.
Theorem 2.5.
For and
one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F579509/MediaObjects/13660_2010_Article_2191_Equ22_HTML.gif)
In (2.19), we can see the relations between the binomial coefficients and the modified extension of the twisted -Bernoulli polynomials of order
References
Jang L-C: Multiple twisted -Euler numbers and polynomials associated with -adic -integrals. Advances in Difference Equations 2008, -11.
Kim T: -Volkenborn integration. Russian Journal of Mathematical Physics 2002, 9(3):288–299.
Kim T: -Bernoulli numbers and polynomials associated with Gaussian binomial coefficients. Russian Journal of Mathematical Physics 2008, 15(1):51–57.
Kim T: Sums of products of -Bernoulli numbers. Archiv der Mathematik 2001, 76(3):190–195. 10.1007/s000130050559
Acknowledgments
The authors would like to thank the anonymous referee for his/her excellent detail comments and suggestions. This Research was supported by Kyungpook National University Research Fund, 2010.
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Rim, SH., Moon, EJ., Lee, SJ. et al. Multivariate Twisted -Adic
-Integral on
Associated with Twisted
-Bernoulli Polynomials and Numbers.
J Inequal Appl 2010, 579509 (2010). https://doi.org/10.1155/2010/579509
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DOI: https://doi.org/10.1155/2010/579509