Volume 2010, Article ID 826548, 8 pagesdoi:10.1155/2010/826548 Research Article On the Symmetric Properties of Associated with the Twisted Generalized Euler Polynomials of Higher Order 1
Trang 1Volume 2010, Article ID 826548, 8 pages
doi:10.1155/2010/826548
Research Article
On the Symmetric Properties of
Associated with the Twisted Generalized Euler
Polynomials of Higher Order
1 Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
2 Department of Wireless Communications Engineering, Kwangwoon University,
Seoul 139-701, Republic of Korea
Correspondence should be addressed to Young-Hee Kim,yhkim@kw.ac.kr
Received 6 November 2009; Revised 11 March 2010; Accepted 14 March 2010
Academic Editor: Ulrich Abel
Copyrightq 2010 Taekyun Kim et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited
We study the symmetric properties for the multivariate p-adic invariant integral onZprelated to the twisted generalized Euler polynomials of higher order
1 Introduction
the ring of rational integers, the ring of p-adic integers, the field of p-adic rational numbers,
invariant integral onZpis defined as
I
f
Zp
N → ∞
pN−1
x0
f xμx p NZp
lim
N → ∞
pN−1
x0
see 1 25 For n ∈ N, we note that
Z f xdμx n−1
l0
−1n−1−l f l see 5. 1.2
Trang 2Let d be a fixed odd positive integer For N ∈ N, we set
X X d lim←−
N
X∗
0<a<dp
a,p1
a dpZ p
,
a dp NZp x ∈ X | x ≡ a
1.3
where a ∈ Z lies in 0 ≤ a < dp Nsee 1 13 It is well known that for f ∈ UDZ p,
X
Zp
For n ∈ N, let C p n be the cyclic group of order p n That is, C p n {ξ | ξ p n
1} The p-adic locally constant space, T p , is defined by T p limn → ∞ C p n n≥1 C p n
t d−1
a0 χ aξ a e at
ξ d e dt− 1 e xt
∞
n0
B n,χ,ξ x t n
In4,7,10–12, the generalized twisted Bernoulli polynomials of order k attached to χ
are also defined as follows:
t d−1
a0 χ aξ a e at
ξ d e dt− 1
× · · · × t
a0 χ aξ a e at
ξ d e dt− 1
k-times
e xt∞
n0
B k n,χ,ξ x t n
Recently, the symmetry identities for the generalized twisted Bernoulli polynomials
In this paper, we study the symmetric properties of the multivariate p-adic invariant
between the power sum polynomials and the generalized higher-order Euler polynomials The main purpose of this paper is to give the symmetry identities for the twisted generalized
Euler polynomials of higher order using the symmetric properties of the multivariate p-adic
invariant integral onZp
Trang 32 Symmetry Identities for the Twisted Generalized
Euler Polynomials of Higher Order
Let χ be Dirichlet’s character with an odd conductor d ∈ N That is, d ∈ N with d ≡ 1 mod 2 For ξ ∈ T p , the twisted generalized Euler polynomials attached to χ, E n,χ,ξ x, are defined as
X
χ
y
ξ y e xyt dμ
y
a0−1a χ aξ a e at
ξ d e dt 1 e xt
∞
n0
E n,χ,ξ x t n
In the special case x 0, E n,χ,ξ E n,χ,ξ 0 are called the nth twisted generalized Euler numbers attached to χ.
From2.1, we note that
X
χ
y
ξ y
x ym
dμ
y
E m,χ,ξ x, m ∈ N ∪ {0}. 2.2
For n ∈ N with n ≡ 1 mod 2, we have
X
χ xξ x e xndt dμ x
X
χ xξ x e xt dμ x 2 nd−1
l0
−1l
Let T k,χ,ξ n n
l0−1l χlξ l l k Then we see that
ξ nd
X
χ xξ x e xndt dμ x
X
χ xξ x e xt dμ x
X e xt χ xξ x dμ x
X e ndxt ξ ndx dμ x 2
∞
k0
T k,χ,ξ nd − 1 t k
k! .
2.4
to χ as follows:
e xt 2
a0−1a
χaξ a e at
ξ d e dt 1
k
∞
n0
E k n,χ,ξ x t n
In the special case x 0, E k n,χ,ξ E k n,χ,ξ 0 are called the nth twisted generalized Euler numbers
of order k.
Trang 4Let w1, w2, d ∈ N with w1≡ 1, w2 ≡ 1, and d ≡ 1 mod 2 Then we set
J χ,ξ m w1, w2| x X m
i1 χ x iξ m i1 x i w1e m i1 x i w2xw1t dμ x1 · · · dμx m
X ξ dw1w2x e dw1w2xt dμ x
×
X m
m
i1
χ x i
ξ m i1 x i w2e m i1 x i w1yw2t dμ x1 · · · dμx m
,
2.6
where
X m f x1, , x m dμx1 · · · dμx m
X
· · ·
X
m-times
f x1, , x m dμx1 · · · dμx m . 2.7
From2.6, we note that
J χ,ξ m w1, w2 | x
X m
m
i1
χ x i
ξ m i1 x i w1e m i1 x i w1t dμ x1 · · · dμx m
e w1w2xt
× X χ x m ξ w2x m e w2x m t dμ x m
X ξ dw1w2x e dw1w2xt dμ x
e w1w2yt
×
X m−1
m−1
i1
χ x i
ξ m−1 i1 x i w2e m−1 i1 x i w2t dμ x1 · · · dμx m−1
.
2.8
X χ xξ x e xt dμ x
X ξ dw1x e dw1xt dμ x
dw1 −1
l0
−1l χ lξ l e lt∞
k0
T k,χ,ξ dw1− 1t k
It is not difficult to show that
e w1w2xt
X m
m
i1
χ x i
ξ m i1 x i w1e m i1 x i w1t dμ x1 · · · dμx m
a0−1a
χaξ aw1e aw1t
ξ dw1e dw1t 1
e w1w2xt∞
k0
E k,χ,ξ m w1 w2xw1k t k
k! .
2.10
Trang 5By2.8, 2.9, and 2.10, we see that
J χ,ξ m w1, w2| x ∞
l0
E m l,χ,ξ w1 w2xw l1t l
l!
∞
k0
T k,χ,ξ w2 w1d − 1w k2t k
k!
i0
E i,χ,ξ m−1 w2
w1y w i
2t i
i!
∞
n0
⎛
j0
n j
w2j w n−j1 E n−j,χ,ξ m w1 w2x
×
j
k0
j k
T k,χ,ξ w2 w1d − 1 E m−1 j−k,χ,ξ w2
w1yt n
n! .
2.11
In the viewpoint of the symmetry of J χ,ξ m w1, w2 | x for w1and w2, we have
J χ,ξ m w1, w2| x ∞
n0
⎛
j0
n j
w1j w n−j2 E n−j,χ,ξ m w2 w1x
k0
j k
T k,χ,ξ w1 w2d − 1 E m−1 j−k,χ,ξ w1
w2yt n
n! .
2.12
theorem
Theorem 2.1 Let w1, w2, d ∈ N with w1 ≡ 1, w2 ≡ 1, and d ≡ 1 mod 2 For n ∈ N ∪ {0} and
m ∈ N, one has
n
j0
n
j
w2j w n−j1 E m n−j,χ,ξ w1 w2x
j
k0
j k
T k,χ,ξ w2 w1d − 1 E m−1 j−k,χ,ξ w2w1y
n
j0
n j
w1j w n−j2 E m n−j,χ,ξ w2 w1x
j
k0
j k
T k,χ,ξ w1 w2d − 1 E m−1 j−k,χ,ξ w1
w2y
.
2.13
Corollary 2.2 For w1, w2, d ∈ N with w1 ≡ 1, w2≡ 1, and d ≡ 1 mod 2, one has
n
m0
n m
E m,χ,ξ w1 w2x w m
1w n−m2 T n−m,χ,ξ w2 w1d − 1
n
m0
n m
E m,χ,ξ w2 w1x w n−m
1 w2m T n−m,χ,ξ w1 w2d − 1 see 2.
2.14
Trang 6Let χ be the trivial character and d 1 Then we also have the following corollary.
Corollary 2.3 Let w1, w2 ∈ N with w1 ≡ 1, w2≡ 1 mod 2 Then one has
n
j0
n j
w n−j1 w j2E n−j,ξ w1 w2x T k,ξ w2 w1− 1
n
j0
n j
w j1w n−j2 E n−j,ξ w2 w1x T k,ξ w1 w2− 1,
2.15
where E n,ξ x are the nth twisted Euler polynomials.
Corollary 2.4 Distribution for the twisted Euler polynomials For w1∈ N with w1≡ 1 mod 2, one has
E n,ξ x n
i0
n i
w i1E i,ξ w1 xT n−i,ξ w1− 1. 2.16 From2.6, we can derive that
J χ,ξ m w1, w2| x
w1d−1
l0
χ l−1 l ξ w2l
X m
m
i1
χ x i
×ξ m i1 x i w1e w1 m
i1 x i w2/w1lw2xt dμ x1 · · · dμx m
×
X m−1
m−1
i1
χ x i
ξ m−1 i1 x i w2e m−1 i1 x i w2t dμ x1 · · · dμx m−1
∞
n0
n
k0
n k
w k1w n−k2 E m−1 n−k,χ,ξ w2
w1y
×w1d−1
l0
χ l−1 l ξ w2l E m k,χ,ξ w1
w2x w2
w1l
t n
n! .
2.17
By the symmetry property of J χ,ξ m w1, w2 | x in w1and w2, we also see that
J χ,ξ m w1, w2 | x
∞
n0
n
k0
n k
w2k w n−k1 E n−k,χ,ξ m−1 w1
w2yw2d−1 l0
χ l−1 l ξ w1l E m k,χ,ξ w2
w1x w1
w2l
t n
n! .
2.18
Trang 7Comparing the coefficients on both sides of 2.17 and 2.18, we obtain the following theorem which shows the relationship between the power sums and the twisted generalized Euler polynomials of higher order
Theorem 2.5 Let w1, w2, d ∈ N with w1 ≡ 1, w2 ≡ 1, and d ≡ 1 mod 2 For n ∈ N ∪ {0} and
m ∈ N, one has
n
k0
n
k
w k1w n−k2 E m−1 n−k,χ,ξ w2
w1yw1d−1 l0
χ l−1 l
ξ w2l E m k,χ,ξ w1
w2x w2
w1l
n
k0
n k
w k
2w n−k
1 E m−1 n−k,χ,ξ w1
w2yw2d−1 l0
χ l−1 l ξ w1l E m k,χ,ξ w2
w1x w1
w2l
.
2.19
n
k0
n k
E k,χ,ξ w1 w k1w n−k2 T n−k,χ,ξ w2 w1d − 1
n
k0
n k
E k,χ,ξ w2 w k2w n−k1 T n−k,χ,ξ w1 w2d − 1 .
2.20
Acknowledgment
The present research has been conducted by the research grant of Kwangwoon University in 2010
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... class="page_container" data-page="7">Comparing the coefficients on both sides of 2.17 and 2.18, we obtain the following theorem which shows the relationship between the power sums and the twisted generalized. .. are the nth twisted Euler polynomials.
Corollary 2.4 Distribution for the twisted Euler polynomials For w1∈ N with w1≡ mod 2, one...
The present research has been conducted by the research grant of Kwangwoon University in 2010
References
1 T Kim and Y.-H Kim, ? ?On the symmetric properties