doi:10.1155/2009/273545
Research Article
On the Identities of Symmetry for the
ζ
-Euler
Polynomials of Higher Order
Taekyun Kim,
1Kyoung Ho Park,
2and Kyung-won Hwang
31Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, South Korea 2Department of Mathematics, Sogang University, Seoul 121-742, South Korea
3Department of General Education, Kookmin University, Seoul 139-702, South Korea
Correspondence should be addressed to Taekyun Kim,[email protected]
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 multivariatep-adic fermionic integral onZp. 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 theζ -Euler polynomials of higher order, we can obtain many identities related to the Frobenius--Euler polynomials of higher order.
Copyrightq2009 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
Letpbe a fixed odd prime number. Throughout this paper,Zp,Qp,C,andCpwill, respectively,
denote the ring ofp-adic rational integer, the field ofp-adic rational numbers, the complex number field, and the completion of algebraic closure of Qp. Let vp be the normalized
exponential valuation ofCpwith|p|p p−vpp p−1. LetUDZpbe the space of uniformly
differentiable functions onZp. Forf ∈UDZp,q∈Cpwith|1−q|p<1, the fermionicp-adic
q-integral onZpis defined as
I−q
f
Zp
fxdμ−qx lim N→ ∞
1q
1qpN
pN−1
x0
fx−qx
1.1
see1. Let us define the fermionicp-adic invariant integral onZpas follows:
I−1
flim
q→1I−q
f
Zp
see1–8. From1.2, we have
I−1
f1
I−1
f2f0 1.3
see9,10, wheref1x fx1. Forζ∈Cpwith|1−ζ|p <1, letfx extζx. Then, we
define theζ-Euler numbers as follows:
Zp
ζxextdμ
−1x
2
ζet1
∞
n0 En,ζt
n
n!, 1.4
whereEn,ζare called theζ-Euler numbers. We can show that
2
ζet1
1ζ−1 etζ−1 ·
2 1ζ
2 1ζ
∞
n0 Hn
−ζ−1tn
n!, 1.5
whereHn−ζ−1are the Frobenius-Euler numbers. By comparing the coefficients on both sides
of1.4and1.5, we see that
En,ζ
2 1ζHn
−ζ−1. 1.6
Now, we also define theζ-Euler polynomials as follows:
2
ζet1e xt∞
n0
En,ζxt n
n!. 1.7
In the viewpoint of1.5, we can show that
2
ζet1e
xtext1ζ−1
etζ−1 ·
2 1ζ
2 1ζ
∞
n0 Hn
−ζ−1, xtn
n!, 1.8
whereHn−ζ−1, xare the nth Frobenius-Euler polynomials. From1.7and1.8, we note
that
En,ζx
2 1ζHn
−ζ−1, x 1.9
cf.1–8,11–18. For each positive integerk, letTk,ζn n0−1ζk. Then we have
∞
k0
Tk,ζnt k
k!
∞
k0
n
0
−1kζ
tk
k!
n
0
−1ζet 1 −1n1en1t
Theζ-Euler polynomials of orderk, denotedEn,ζkx, are defined as
ext
2
ζet1 k
2
ζet1
× · · · ×
2
ζet1
ext
∞
n0
En,ζkxt
n
n!. 1.11
Then the values ofEn,ζkxatx 0 are called theζ-Euler numbers of orderk. Whenk 1, the polynomials or numbers are called theζ-Euler polynomials or numbers. The purpose of this paper is to investigate some properties of symmetry for the multivariatep-adic fermionic integral onZp. From the properties of symmetry for the multivariatep-adic fermionic integral
onZp, we derive some identities of symmetry for theζ-Euler polynomials of higher order. By
using our identities of symmetry for theζ-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
Letw1, w2∈Nwithw1≡1mod 2andw2 ≡1mod 2. Then we set
Rmw1, w2
Zm
pe
w1x1x2···xmw2xtζw1x1···w1xmdμ−
1x1· · ·dμ−1xm
Zpζ
w1w2xew1w2xtdμ−1x
×
Zm p
ew2x1x2···xmw1ytζw2x1···w2xmdμ
−1x1· · ·dμ−1xm,
2.1
where
Zm p
fx1, . . . , xmdμ−1x1· · ·dμ−1xm
Zp · · ·
Zp
fx1, . . . , xmdμ−1x1· · ·dμ−1xm. 2.2
Thus, we note that this expression forRmw
1, w2is symmetry inw1andw2. From2.1, we
have
Rmw1, w2
Zm p
ew1x1···xmtζw1x1···w1xmdμ
−1x1· · ·dμ−1xm
ew1w2xt
×
⎛ ⎝
Zpe
w2xmtζw2xmdμ
−1xm
Zpe
w1w2xtζw1w2xdμ−1x
⎞ ⎠
×
Zm−1 p
ew2x1···xm−1tζw2x1···w2xm−1dμ
−1x1· · ·dμ−1xm−1
We can show that
Thus, we have
Lety0 andm1 in2.9. Then we have
n
j0 n j
wn1−jwj2En−j,ζw1w2xTk,ζw2w1−1
n
j0 n j
wj1wn2−jEn−j,ζw2w1xTk,ζw1w2−1.
2.10
From2.10, we note that
n
i0 n
i
wi1w2n−iEi,ζw1w2xTn−i,ζw2w1−1
n
i0 n
i
wn1−iw2iEi,ζw2w1xTn−i,ζw1w2−1.
2.11
If we takew21 in2.11, then we have
En,ζw1x
n
i0 n
i
w1iEi,ζw1xTn−i,ζw1−1. 2.12
From2.3, we note that
Rmw1, w2
Zm p
ew1x1···xmtζw1x1···w1xmdμ
−1x1· · ·dμ−1xm
ew1w2xt
×
⎛ ⎝
Zpe
w2xmtζw2xmdμ−1xm
Zpe
w1w2xtζw1w2xdμ−1x
⎞ ⎠
×
Zm−1 p
ew2x1···xm−1tζw2x1···w2xm−1dμ
−1x1· · ·dμ−1xm−1
ew1w2yt
Zm p
ew1x1···xmtζw1x1···w1xmdμ
−1x1· · ·dμ−1xm
ew1w2xt
× w1−1
i0
−1iew2itζw2i
×
Zm−1 p
ew2x1···xm−1tζw2x1···w2xm−1dμ
−1x1· · ·dμ−1xm−1
∞
n0
n
k0
w2−1
i0
−1iζw1iEm
k,ζw2
w1x w1 w2i
wk
2 k!E
m−1 n−k
w2y
wn1−k n−k!n!
tn
n!
∞
n0
n
k0 n k
w2kw1n−kEnm−k,ζ−1w1
w2y
w2−1
i0
−1iζw1iEm
k,ζw2
w1xw1 w2i
tn
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∈Nwithw1≡1mod 2 and w2≡1mod 2, one has
n
k0 n k
w1kw2n−kEnm−k,ζ−1w2
w1y
w1−1
i0
−1iζw2iEm
k,ζw1
w2xw2 w1i
n
k0 n k
w2kwn1−kEnm−k,ζ−1w1
w2y
w2−1
i0
−1iζw1iEm
k,ζw2
w1xw1 w2i
.
2.15
Lety0 andm1, we have
w1n
w1−1
i0
−1iζw2iE
n,ζw1
w2xw2 w1i
wn2
w2−1
i0
−1iζw1iE
n,ζw2
w1xw1 w2i
. 2.16
From2.16, we can derive
w1−1
i0
−1iζiEn,ζw1
x 1 w1i
1 wn
1
En,ζw1x. 2.17
Acknowledgment
The present research has been conducted by the research grant of the Kwangwoon University in 2009.
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