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Symmetry Properties of Higher-Order Bernoulli Polynomials
Advances in Difference Equations volume 2009, Article number: 318639 (2009)
Abstract
We investigate properties of identities and some interesting identities of symmetry for the Bernoulli polynomials of higher order using the multivariate -adic invariant integral on
.
1. Introduction
Let be a fixed prime number. Throughout this paper
and
will, respectively, denote the ring of
-adic rational integers, the field of
-adic rational numbers, and the completion of algebraic closure of
. For
, we use the notation
. Let
be the space of uniformly differentiable functions on
and let
be the normalized exponential valuation of
with
. For
with
, the
-Volkenborn integral on
is defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F318639/MediaObjects/13662_2009_Article_1183_Equ1_HTML.gif)
(see [1, 2]). The ordinary -adic invariant integral on
is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F318639/MediaObjects/13662_2009_Article_1183_Equ2_HTML.gif)
(see [1–15]). Let . Then we easily see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F318639/MediaObjects/13662_2009_Article_1183_Equ3_HTML.gif)
From (1.3), we can derive
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F318639/MediaObjects/13662_2009_Article_1183_Equ4_HTML.gif)
(see [2, 8–10]), where are the
th Bernoulli numbers.
By (1.2) and (1.3), we easily see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F318639/MediaObjects/13662_2009_Article_1183_Equ5_HTML.gif)
where for
It is known that the Bernoulli polynomials are defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F318639/MediaObjects/13662_2009_Article_1183_Equ6_HTML.gif)
where are called the
th Bernoulli polynomials. The Bernoulli polynomials of order
, denoted
, are defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F318639/MediaObjects/13662_2009_Article_1183_Equ7_HTML.gif)
(see [3–6]). Then the values of at
are called the Bernoulli numbers of order
. When
, the polynomials or numbers are called the Bernoulli polynomials or numbers. The purpose of this paper is to investigate some interesting properties of symmetry for the multivariate
-adic invariant integral on
. From the properties of symmetry for the multivariate
-adic invariant integral on
, we derive some interesting identities of symmetry for the Bernoulli polynomials of higher order.
2. Symmetry Properties of Higher-Order Bernoulli Polynomials
Let . Then we define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F318639/MediaObjects/13662_2009_Article_1183_Equ8_HTML.gif)
From (2.1), we note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F318639/MediaObjects/13662_2009_Article_1183_Equ9_HTML.gif)
where
In (2.1), we note that is symmetric in
. By (2.1), we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F318639/MediaObjects/13662_2009_Article_1183_Equ10_HTML.gif)
It is easy to see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F318639/MediaObjects/13662_2009_Article_1183_Equ11_HTML.gif)
From (2.1), (2.3), and the above formula, we can derive
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F318639/MediaObjects/13662_2009_Article_1183_Equ12_HTML.gif)
By the symmetry of in
and
, we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F318639/MediaObjects/13662_2009_Article_1183_Equ13_HTML.gif)
By comparing the coefficients on both sides of (2.5) and (2.6), we obtain the following theorem.
Theorem 2.1.
For , one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F318639/MediaObjects/13662_2009_Article_1183_Equ14_HTML.gif)
Let and
in (2.7). Then we have the following corollary.
Corollary 2.2.
For , one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F318639/MediaObjects/13662_2009_Article_1183_Equ15_HTML.gif)
If we take in (2.8), then we also obtain the following corollary.
Corollary 2.3.
For one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F318639/MediaObjects/13662_2009_Article_1183_Equ16_HTML.gif)
By the definition of , we easily see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F318639/MediaObjects/13662_2009_Article_1183_Equ17_HTML.gif)
From the symmetric property of in
, we note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F318639/MediaObjects/13662_2009_Article_1183_Equ18_HTML.gif)
By comparing the coefficients on both sides of (2.10) and (2.11), we obtain the following theorem.
Theorem 2.4.
For , one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F318639/MediaObjects/13662_2009_Article_1183_Equ19_HTML.gif)
Let and
in (2.12). Then we obtain the following Corollary 2.5.
Corollary 2.5.
For , one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F318639/MediaObjects/13662_2009_Article_1183_Equ20_HTML.gif)
From (2.12), we can get the well-known result due to Raabe:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F318639/MediaObjects/13662_2009_Article_1183_Equ21_HTML.gif)
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Kim, T., Hwang, KW. & Kim, YH. Symmetry Properties of Higher-Order Bernoulli Polynomials. Adv Differ Equ 2009, 318639 (2009). https://doi.org/10.1155/2009/318639
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DOI: https://doi.org/10.1155/2009/318639