Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2009, Article ID 273545, 9 pages
doi:10.1155/2009/273545
Research Article
On the Identities of Symmetry for the ζ-Euler
Polynomials of Higher Order
Taekyun Kim,1 Kyoung Ho Park,2 and Kyung-won Hwang3
1
Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, South Korea
Department of Mathematics, Sogang University, Seoul 121-742, South Korea
3
Department of General Education, Kookmin University, Seoul 139-702, South Korea
2
Correspondence should be addressed to Taekyun Kim, tkkim@kw.ac.kr
Received 19 February 2009; Revised 31 May 2009; Accepted 18 June 2009
Recommended by Agacik Zafer
The main purpose of this paper is to investigate several further interesting properties of symmetry
for the multivariate p-adic fermionic integral on Zp . From these symmetries, we can derive some
recurrence identities for the ζ-Euler polynomials of higher order, which are closely related to
the Frobenius-Euler polynomials of higher order. By using our identities of symmetry for the ζEuler polynomials of higher order, we can obtain many identities related to the Frobenius-Euler
polynomials of higher order.
Copyright q 2009 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.
1. Introduction/Definition
Let p be a fixed odd prime number. Throughout this paper, Zp , Qp , C, and Cp will, respectively,
denote the ring of p-adic rational integer, the field of p-adic rational numbers, the complex
number field, and the completion of algebraic closure of Qp . Let vp be the normalized
exponential valuation of Cp with |p|p p−vp p p−1 . Let UDZp be the space of uniformly
differentiable functions on Zp . For f ∈ UDZp , q ∈ Cp with |1 − q|p < 1, the fermionic p-adic
q-integral on Zp is defined as
I−q f −1
x
1 q p
fxdμ−q x lim
fx −q
N
p
N →∞1 q
Zp
x0
N
1.1
see 1. Let us define the fermionic p-adic invariant integral on Zp as follows:
I−1 f lim I−q f q→1
Zp
fxdμ−1 x
1.2
2
Advances in Difference Equations
see 1–8. From 1.2, we have
I−1 f1 I−1 f 2f0
1.3
see 9, 10, where f1 x fx 1. For ζ ∈ Cp with |1 − ζ|p < 1, let fx ext ζx . Then, we
define the ζ-Euler numbers as follows:
Zp
ζx ext dμ−1 x ∞
tn
2
,
E
n,ζ
n!
ζet 1 n0
1.4
where En,ζ are called the ζ-Euler numbers. We can show that
∞
n
2
2 1 ζ−1
2
−1 t
,
−ζ
·
H
n
n!
ζet 1 et ζ−1 1 ζ 1 ζ n0
1.5
where Hn −ζ−1 are the Frobenius-Euler numbers. By comparing the coefficients on both sides
of 1.4 and 1.5, we see that
En,ζ 2
Hn −ζ−1 .
1ζ
1.6
Now, we also define the ζ-Euler polynomials as follows:
∞
tn
2
xt
e
En,ζ x .
t
n!
ζe 1
n0
1.7
In the viewpoint of 1.5, we can show that
∞
tn
−1
2
2 2
xt
xt 1 ζ
−1
−ζ
,
e
e
·
H
,
x
n
n!
ζet 1
et ζ−1 1 ζ 1 ζ n0
1.8
where Hn −ζ−1 , x are the nth Frobenius-Euler polynomials. From 1.7 and 1.8, we note
that
En,ζ x 2
Hn −ζ−1 , x
1ζ
cf. 1–8, 11–18. For each positive integer k, let Tk,ζ n 1.9
n
k
0 −1 ζ .
Then we have
∞
n
n
tk tk 1 −1n1 en1t
k .
Tk,ζ n −1 ζ
−1 ζ et k! k0 0
k! 0
ζet 1
k0
∞
1.10
Advances in Difference Equations
3
k
The ζ-Euler polynomials of order k, denoted En,ζ x, are defined as
e
xt
2
ζet 1
k
2
ζet 1
× ··· ×
∞
tn
2
k
xt
.
e
E
x
n,ζ
n!
ζet 1
n0
1.11
k
Then the values of En,ζ x at x 0 are called the ζ-Euler numbers of order k. When k 1,
the polynomials or numbers are called the ζ-Euler polynomials or numbers. The purpose of
this paper is to investigate some properties of symmetry for the multivariate p-adic fermionic
integral on Zp . From the properties of symmetry for the multivariate p-adic fermionic integral
on Zp , we derive some identities of symmetry for the ζ-Euler polynomials of higher order. By
using our identities of symmetry for the ζ-Euler polynomials of higher order, we can obtain
many identities related to the Frobenius-Euler polynomials of higher order.
2. On the Symmetry for the ζ-Euler Polynomials of Higher Order
Let w1 , w2 ∈ N with w1 ≡ 1mod 2 and w2 ≡ 1mod 2. Then we set
m
R
Zm
p
w1 , w2 ×
ew1 x1 x2 ···xm w2 xt ζw1 x1 ···w1 xm dμ−1 x1 · · · dμ−1 xm ζw1 w2 x ew1 w2 xt dμ−1 x
Zp
Zm
p
2.1
ew2 x1 x2 ···xm w1 yt ζw2 x1 ···w2 xm dμ−1 x1 · · · dμ−1 xm ,
where
Zm
p
fx1 , . . . , xm dμ−1 x1 · · · dμ−1 xm Zp
···
Zp
fx1 , . . . , xm dμ−1 x1 · · · dμ−1 xm . 2.2
Thus, we note that this expression for Rm w1 , w2 is symmetry in w1 and w2 . From 2.1, we
have
m
R
w1 , w2 Zm
p
e
w1 x1 ···xm t w1 x1 ···w1 xm
⎛
× ⎝
Zp
ζ
ew2 xm t ζw2 xm dμ−1 xm ew1 w2 xt ζw1 w2 x dμ−1 x
Zp
⎞
⎠
2.3
×
dμ−1 x1 · · · dμ−1 xm ew1 w2 xt
Zm−1
p
e
w2 x1 ···xm−1 t w2 x1 ···w2 xm−1
ζ
dμ−1 x1 · · · dμ−1 xm−1 ew1 w2 yt .
4
Advances in Difference Equations
We can show that
Zp
Zp
ext ζx dμ−1 x
ew1 xt ζw1 x dμ−1 x
w
1 −1
−1 ζ et 0
∞
Tk,ζ w1 − 1
k0
tk
.
k!
2.4
By 1.4 and 1.11, we see that
Zm
p
e
w1 x1 ···xm t w1 x1 ···w1 xm
ζ
2
m
ζ w1 e w1 t 1
dμ−1 x1 · · · dμ−1 xm ew1 w2 xt
2.5
ew1 w2 xt
∞
tn
m
En,ζw1 w2 xw1n .
n!
n0
Thus, we have
m
En,ζw1 w2 x
n
n
0
m
2.6
E,ζw1 w2n− xn− .
From 2.3, 2.4, and 2.5, we can derive
m
R
w1 , w2 ∞
t
m
E,ζw1 w2 xw1
!
0
∞
wk
Tk,ζw2 w1 − 1 2 tk
k!
k0
∞
i0
wi
m−1 Ei,ζw2 w1 y 2 ti
i!
⎞ ⎞
⎛ ⎛ j
m−1 ∞
∞
Ej−k w1 y
tj
j−k
m
t
k
⎝ ⎝ Tk,ζw2 w1 − 1w w
E,ζw1 w2 xw1
j!⎠ ⎠
2 2
!
j!
k! j − k !
j0
0
k0
⎞
⎛ j
∞
∞
j
tj
t ⎝
j
m
m−1
E,ζw1 w2 xw1
Tk,ζw2 w1 − 1
Ej−k,ζw2 w1 y w2 ⎠
!
j!
k
j0 k0
0
⎛ ⎞
j n−j
j
∞
n
j
w
w
tn
m−1
m
⎝
Ej−k,ζw2 w1 y
Tk,ζw2 w1 − 1
2 1 En−j,ζw1 w2 xn!⎠
n!
n − j !j!
k
n0
j0 k0
⎞
⎛ j
∞
n
j
n
tn
j
n−j
m
m−1
⎝
Ej−k,ζw2 w1 y ⎠ .
w2 w1 En−j,ζw1 w2 x Tk,ζw2 w1 − 1
n!
k
j
n0
j0
k0
2.7
Advances in Difference Equations
5
By the same method, we also see that
m
R
w1 , w2 e
Zm
p
w2 x1 ···xm t w2 x1 ···w2 xm
ζ
⎛
× ⎝
Zp
ew1 xm t ζw1 xm dμ−1 xm Zp
ew1 w2 xt ζw1 w2 x dμ−1 x
⎞
⎠
×
dμ−1 x1 · · · dμ−1 xm ew1 w2 xt
Zm−1
p
e
w1 x1 ···xm−1 t w1 x1 ···w1 xm−1
ζ
dμ−1 x1 · · · dμ−1 xm−1 ew1 w2 yt
∞
∞
∞
i ti
tk
m
m−1 t
k
E,ζw2 w1 xw2
Tk,ζw1 w2 − 1w1
Ei,ζw1 w2 y w1
!
k!
i!
i0
0
k0
⎞
⎞
⎛ ⎛ j
m−1 ∞
∞
y
w
2
w1 w2 − 1 Ej−k
T
t
k,ζ
j
m
⎝ ⎝
E,ζw2 w1 xw2
⎠ w 1 tj ⎠
!
k!
j
−
k
!
j0
0
k0
⎞ j ⎞
⎛ ⎛ j
m−1 ∞
∞
Tk,ζw1 w2 − 1Ej−k w2 y
w tj
m
t
⎝ ⎝
j!⎠ 1 ⎠
E,ζw2 w1 xw2
!
j!
k! j − k !
j0
0
k0
∞
t
m
E,ζw2 w1 xw2
!
0
⎞
⎛ j ∞
j
j
jt
m−1
⎝
Tk,ζw1 w2 − 1Ej−k,ζw1 w2 y w1 ⎠
j!
k
j0 k0
⎞
⎛ j n−j
j
∞
n
j
w1 w2
tn
m−1
m
⎝
Tk,ζw1 w2 − 1Ej−k,ζw1 w2 y
En−j,ζw2 w1 xn!⎠
n!
j! n − j !
n0
j0 k0 k
⎛ ⎞
j
∞
n
n
j
tn
j
n−j
m
m−1
⎝
w1 w2 En−j,ζw2 w1 x
Tk,ζw1 w2 − 1Ej−k,ζw1 w2 y ⎠ .
n!
j
n0
j0
k0 k
2.8
By comparing the coefficients on both sides of 2.7 and 2.8, we obtain the following.
Theorem 2.1. For w1 , w2 ∈ N with w1 ≡ 1mod 2, w2 ≡ 1mod 2, and n ≥ 0, m ≥ 1, one has
n
n
j0
j
j
j n−j m
w2 w1 En−j,ζw1 w2 x Tk,ζw2 w1
k0
n
n
j0
j
j
j n−j m
w1 w2 En−j,ζw2 w1 x
k0
j
m−1 Ej−k,ζw2 w1 y
− 1
k
j
m−1 Tk,ζw1 w2 − 1Ej−k,ζw1 w2 y .
k
2.9
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Advances in Difference Equations
Let y 0 and m 1 in 2.9. Then we have
n
n
j
j0
n−j
j
w1 w2 En−j,ζw1 w2 xTk,ζw2 w1 − 1
2.10
n
n
j0
j
j n−j
w1 w2 En−j,ζw2 w1 xTk,ζw1 w2
− 1.
From 2.10, we note that
n
n
i
i0
w1i w2n−i Ei,ζw1 w2 xTn−i,ζw2 w1 − 1
2.11
n
n
i0
i
w1n−i w2i Ei,ζw2 w1 xTn−i,ζw1 w2 − 1.
If we take w2 1 in 2.11, then we have
En,ζ w1 x n
n
i0
i
w1i Ei,ζw1 xTn−i,ζ w1 − 1.
2.12
From 2.3, we note that
m
R
w1 , w2 Zm
p
e
w1 x1 ···xm t w1 x1 ···w1 xm
ζ
⎛
× ⎝
Zp
ew2 xm t ζw2 xm dμ−1 xm Zp
ew1 w2 xt ζw1 w2 x dμ−1 x
Zm−1
p
⎞
⎠
×
dμ−1 x1 · · · dμ−1 xm ew1 w2 xt
Zm
p
e
e
w2 x1 ···xm−1 t w2 x1 ···w2 xm−1
ζ
dμ−1 x1 · · · dμ−1 xm−1 ew1 w2 yt
w1 x1 ···xm t w1 x1 ···w1 xm
ζ
dμ−1 x1 · · · dμ−1 xm ew1 w2 xt
w −1
1
i w2 it w2 i
×
−1 e ζ
i0
×
Zm−1
p
e
w2 x1 ···xm−1 t w2 x1 ···w2 xm−1
ζ
dμ−1 x1 · · · dμ−1 xm−1 ew1 w2 yt
Advances in Difference Equations
7
w −1
1
i w2 i
w1 x1 ···xm w2 /w1 iw2 xt w1 x1 ···w1 xm
e
ζ
dμ−1 x1 · · · dμ−1 xm −1 ζ
Zm
p
i0
×
Zm−1
p
e
w2 x1 ···xm−1 w1 yt w2 x1 ···w2 xm−1
ζ
dμ−1 x1 · · · dμ−1 xm−1 w −1
∞
∞
k
1
t
w2
m
m−1 i w2 i
kt
Ek,ζw1
i w2 x w 1
E,ζw2 w1 y w2
−1 ζ
w1
k!
!
i0
k0
0
w −1
k
∞
n
1
w1 m−1 w2n−k
tn
w2
i w2 i m
En−k,ζw2 w1 y
n!
i
−1 ζ Ek,ζw1 w2 x w1
k!
n!
n − k!
n0 k0
i0
∞
n
n
n0
m−1 w1k w2n−k En−k,ζw2 w1 y
k
k0
w
1 −1
i w2 i
−1 ζ
i0
m
Ek,ζw1
n
t
w2
.
i
w2 x w1
n!
2.13
By the symmetric property of Rm w1 , w2 in w1 , w2 , we also see that
m
R
w1 , w2 Zm
p
e
w2 x1 ···xm t w2 x1 ···xm ζ
⎛
× ⎝
Zp
ew1 xm t ζw1 xm dμ−1 xm Zp
ew1 w2 xt ζw1 w2 x dμ−1 x
Zm−1
p
⎞
⎠
×
dμ−1 x1 · · · dμ−1 xm ew1 w2 xt
Zm
p
e
e
w1 x1 ···xm−1 t w1 x1 ···xm−1 ζ
dμ−1 x1 · · · dμ−1 xm−1 ew1 w2 yt
w2 x1 ···xm t w2 x1 ···xm ζ
dμ−1 x1 · · · dμ−1 xm ew1 w2 xt
w −1
2
i w1 it w1 i
×
−1 e ζ
i0
×
Zm−1
p
e
w1 x1 ···xm−1 w2 yt w1 x1 ···xm−1 ζ
i w1 i
−1 ζ
i0
×
w
2 −1
Zm
p
Zm−1
p
e
dμ−1 x1 · · · dμ−1 xm−1 e
w2 x1 ···xm w1 /w2 iw1 xt w2 x1 ···xm ζ
dμ−1 x1 · · · dμ−1 xm w1 x1 ···xm−1 w2 yt w1 x1 ···w1 xm−1
ζ
dμ−1 x1 · · · dμ−1 xm−1 w −1
∞
∞
k
2
t
w1
m
m−1 i w1 i
kt
Ek,ζw2
i w1 x w 2
E,ζw1 w2 y w1
−1 ζ
w2
k!
!
i0
k0
0
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Advances in Difference Equations
w −1
k
∞
n
2
w2 m−1 w1n−k
w1
tn
i w1 i m
En−k w2 y
i
n!
−1 ζ Ek,ζw2 w1 x w2
k!
n!
n − k!
n0 k0
i0
∞
n
n
n0
k
k0
m−1 w2k w1n−k En−k,ζw1 w2 y
w
2 −1
i w1 i
−1 ζ
i0
m
Ek,ζw2
n
t
w1
.
i
w1 x w2
n!
2.14
By comparing the coefficients on both sides of 2.13 and 2.14, we obtain the following
theorem.
Theorem 2.2. For w1 , w2 ∈ N with w1 ≡ 1mod 2 and w2 ≡ 1mod 2 , one has
n
n
k0
m−1 w1k w2n−k En−k,ζw2 w1 y
k
w
1 −1
−1 ζ
i0
n
n
m−1 w2k w1n−k En−k,ζw1 w2 y
k
k0
i w2 i
w
2 −1
m
Ek,ζw1
i w1 i
−1 ζ
i0
w2
w2 x i
w1
m
Ek,ζw2
w1
i .
w1 x w2
2.15
Let y 0 and m 1, we have
w1n
w
1 −1
i w2 i
−1 ζ
E
n,ζw1
i0
w2
i
w2 x w1
w2n
w
2 −1
i w1 i
−1 ζ
i0
E
n,ζw2
w1
i .
w1 x w2
2.16
From 2.16, we can derive
w
1 −1
i i
−1 ζ E
i0
n,ζw1
1
i
x
w1
1
En,ζ w1 x.
w1n
2.17
Acknowledgment
The present research has been conducted by the research grant of the Kwangwoon University
in 2009.
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