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On the Identities of Symmetry for the Generalized Bernoulli Polynomials Attached to
of Higher Order
Journal of Inequalities and Applications volume 2009, Article number: 640152 (2009)
Abstract
We give some interesting relationships between the power sums and the generalized Bernoulli numbers attached to of higher order using multivariate
-adic invariant integral on
.
1. Introduction
Let be a fixed prime number. Throughout this paper, the symbols
,
,
, and
denote the ring of rational integers, the ring of
-adic integers, the field of
-adic rational numbers, and the completion of algebraic closure of
, respectively. Let
be the set of natural numbers, and
. Let
be the normalized exponential valuation of
with
(see [1–24]). Let
be the space of uniformly differentiable function on
. Let
be a fixed positive integer. For
, let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F640152/MediaObjects/13660_2009_Article_1985_Equ1_HTML.gif)
where lies in
. For
, the
-adic invariant integral on
is defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F640152/MediaObjects/13660_2009_Article_1985_Equ2_HTML.gif)
(see [11–19]). From (1.2), we note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F640152/MediaObjects/13660_2009_Article_1985_Equ3_HTML.gif)
where and
. Let
. Then we can derive the following equation from (1.3):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F640152/MediaObjects/13660_2009_Article_1985_Equ4_HTML.gif)
(see [1–11]). Let be the Dirichlet's character with conductor
. Then the generalized Bernoulli polynomials attached to
are defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F640152/MediaObjects/13660_2009_Article_1985_Equ5_HTML.gif)
and the generalized Bernoulli numbers attached to ,
, are defined as
(see [1–20, 25]). The purpose of this paper is to derive some identities of symmetry for the generalized Bernoulli polynomials attached to
of higher order.
2. Symmetric Properties for the Generalized Bernoulli Polynomials of Higher Order
Let be the Dirichlet's character with conductor
. Then we note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F640152/MediaObjects/13660_2009_Article_1985_Equ6_HTML.gif)
where are the
th generalized Bernoulli numbers attached to
(see [7, 9, 15, 25]). Now we also see that the generalized Bernoulli polynomials attached to
are given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F640152/MediaObjects/13660_2009_Article_1985_Equ7_HTML.gif)
By (2.1) and (2.2), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F640152/MediaObjects/13660_2009_Article_1985_Equ8_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F640152/MediaObjects/13660_2009_Article_1985_Equ9_HTML.gif)
(see [1–19, 25]). For , we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F640152/MediaObjects/13660_2009_Article_1985_Equ10_HTML.gif)
where . Thus, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F640152/MediaObjects/13660_2009_Article_1985_Equ11_HTML.gif)
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F640152/MediaObjects/13660_2009_Article_1985_Equ12_HTML.gif)
Let us define the -adic function
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F640152/MediaObjects/13660_2009_Article_1985_Equ13_HTML.gif)
(see [25]). By (2.7) and (2.8), we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F640152/MediaObjects/13660_2009_Article_1985_Equ14_HTML.gif)
(see [25]). Thus, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F640152/MediaObjects/13660_2009_Article_1985_Equ15_HTML.gif)
This means that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F640152/MediaObjects/13660_2009_Article_1985_Equ16_HTML.gif)
(see [25]).
The generalized Bernoulli polynomials attached to of order
, which is denoted by
, are defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F640152/MediaObjects/13660_2009_Article_1985_Equ17_HTML.gif)
Then the values of at
are called the generalized Bernoulli numbers attached to
of order
. When
, the polynomials of numbers are called the generalized Bernoulli polynomials or numbers attached to
. Let
. Then we set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F640152/MediaObjects/13660_2009_Article_1985_Equ18_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F640152/MediaObjects/13660_2009_Article_1985_Equ19_HTML.gif)
In (2.13), we note that is symmetric in
. From (2.13), we derive
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F640152/MediaObjects/13660_2009_Article_1985_Equ20_HTML.gif)
It is easy to see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F640152/MediaObjects/13660_2009_Article_1985_Equ21_HTML.gif)
From (2.16), we note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F640152/MediaObjects/13660_2009_Article_1985_Equ22_HTML.gif)
By the symmetry of in
and
, we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F640152/MediaObjects/13660_2009_Article_1985_Equ23_HTML.gif)
By comparing the coefficients on the both sides of (2.17) and (2.18), we see the following theorem.
Theorem 2.1.
For , one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F640152/MediaObjects/13660_2009_Article_1985_Equ24_HTML.gif)
Remark 2.2.
Let and
in (1.4). Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F640152/MediaObjects/13660_2009_Article_1985_Equ25_HTML.gif)
(see [25]).
We also calculate that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F640152/MediaObjects/13660_2009_Article_1985_Equ26_HTML.gif)
From the symmetric property of in
and
, we derive
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F640152/MediaObjects/13660_2009_Article_1985_Equ27_HTML.gif)
By comparing the coefficients on the both sides of (2.21) and (2.22), we obtain the following theorem.
Theorem 2.3.
For , one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F640152/MediaObjects/13660_2009_Article_1985_Equ28_HTML.gif)
Remark 2.4.
Let and
in (2.23). We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F640152/MediaObjects/13660_2009_Article_1985_Equ29_HTML.gif)
(see [25]).
References
Cenkci M: The
-adic generalized twisted
-Euler-
-function and its applications. Advanced Studies in Contemporary Mathematics 2007,15(1):37–47.
Cenkci M, Simsek Y, Kurt V: Multiple two-variable
-adic
-
-function and its behavior at
. Russian Journal of Mathematical Physics 2008,15(4):447–459. 10.1134/S106192080804002X
Jang L-C, Kim S-D, Park D-W, Ro Y-S: A note on Euler number and polynomials. Journal of Inequalities and Applications 2006, 2006:-5.
Kim T:
-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients. Russian Journal of Mathematical Physics 2008,15(1):51–57.
Kim T: A note on
-Volkenborn integration. Proceedings of the Jangjeon Mathematical Society 2005,8(1):13–17.
Kim T:
-Euler numbers and polynomials associated with
-adic
-integrals. Journal of Nonlinear Mathematical Physics 2007,14(1):15–27. 10.2991/jnmp.2007.14.1.3
Kim T: A note on
-adic
-integral on
associated with
-Euler numbers. Advanced Studies in Contemporary Mathematics 2007,15(2):133–137.
Kim T: On the
-extension of Euler and Genocchi numbers. Journal of Mathematical Analysis and Applications 2007,326(2):1458–1465. 10.1016/j.jmaa.2006.03.037
Kim T: On
-adic
-
-functions and sums of powers. Journal of Mathematical Analysis and Applications 2007,329(2):1472–1481. 10.1016/j.jmaa.2006.07.071
Kim T: On the multiple
-Genocchi and Euler numbers. Russian Journal of Mathematical Physics 2008,15(4):481–486. 10.1134/S1061920808040055
Kim T: New approach to
-Euler, Genocchi numbers and their interpolation functions. Advanced Studies in Contemporary Mathematics 2009,18(2):105–112.
Kim T: On a
-analogue of the
-adic log gamma functions and related integrals. Journal of Number Theory 1999,76(2):320–329. 10.1006/jnth.1999.2373
Kim T: Sums of products of -Euler numbers. to appear in Journal of Computational Analysis and Applications to appear in Journal of Computational Analysis and Applications
Kim T, Choi JY, Sug JY: Extended
-Euler numbers and polynomials associated with fermionic
-adic
-integral on
. Russian Journal of Mathematical Physics 2007,14(2):160–163. 10.1134/S1061920807020045
Kim T:
-Volkenborn integration. Russian Journal of Mathematical Physics 2002,9(3):288–299.
Kim Y-H, Kim W, Jang L-C: On the
-extension of Apostol-Euler numbers and polynomials. Abstract and Applied Analysis 2008, 2008:-10.
Pečarić J, Vukelić A: General dual Euler-Simpson formulae. Journal of Mathematical Inequalities 2008,2(4):511–526.
Simsek Y: Complete sum of products of -extension of the Euler polynomials and numbers. http://arxiv.org/abs/0707.2849
Simsek Y: On
-adic twisted
-
- functions related to generalized twisted Bernoulli numbers. Russian Journal of Mathematical Physics 2006,13(3):340–348. 10.1134/S1061920806030095
Simsek Y, Kurt V, Kim D: New approach to the complete sum of products of the twisted
-Bernoulli numbers and polynomials. Journal of Nonlinear Mathematical Physics 2007,14(1):44–56. 10.2991/jnmp.2007.14.1.5
Kim T: Symmetry properties of higher order Bernoulli polynomials. (communicated) (communicated)
Zhang Z, Yang H: Some closed formulas for generalized Bernoulli-Euler numbers and polynomials. Proceedings of the Jangjeon Mathematical Society 2008,11(2):191–198.
Tekcan A, Özkoç A, Gezer B, Bizim O: Some relations involving the sums of Fibonacci numbers. Proceedings of the Jangjeon Mathematical Society 2008,11(1):1–12.
Khrennikov AYu: Generalized probabilities taking values in non-Archimedean fields and in topological groups. Russian Journal of Mathematical Physics 2007,14(2):142–159. 10.1134/S1061920807020033
Kim T, Rim S-H, Lee B: Some identities of symmetry for the generalized Bernoulli numbers and polynomials. Abstract and Applied Analysis 2009, 2009:-8.
Acknowledgment
The present research has been conducted by the research Grant of the Kwangwoon University in 2009.
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Kim, T., Jang, LC., Kim, YH. et al. On the Identities of Symmetry for the Generalized Bernoulli Polynomials Attached to of Higher Order.
J Inequal Appl 2009, 640152 (2009). https://doi.org/10.1155/2009/640152
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DOI: https://doi.org/10.1155/2009/640152