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Volume 2009, Article ID 273545, 9 pagesdoi:10.1155/2009/273545 Research Article Polynomials of Higher Order 1 Division of General Education-Mathematics, Kwangwoon University, Seoul 139-7

Volume 2009, Article ID 273545, 9 pages

doi:10.1155/2009/273545

Research Article

Polynomials of Higher Order

1 Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, South Korea

2 Department of Mathematics, Sogang University, Seoul 121-742, South Korea

3 Department of General Education, Kookmin University, Seoul 139-702, South Korea

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 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 the

ζ-Euler polynomials of higher order, we can obtain many identities related to the Frobenius-ζ-Euler polynomials of higher order

Copyrightq 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 , Q p , C, and C pwill, 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 ofCpwith|p| p  p −v p p  p−1 Let UDZp be the space of uniformly

differentiable functions on Zp For f ∈ UDZp, q ∈ Cpwith|1 − q| p < 1, the fermionic p-adic q-integral on Z pis defined as

I −q

f





Zp

f xdμ −q x  lim

N → ∞

1 q

1 q p N

pN−1

x0

f x−qx 1.1

see 1 Let us define the fermionic p-adic invariant integral on Zpas follows:

I−1

f

 lim

q → 1 I −q

f





Zp

see 1 8 From 1.2, we have

I−1

f1



 I−1

f

see 9,10, where f1x  fx  1 For ζ ∈ Cpwith|1 − ζ| p < 1, let fx  e xt ζ x Then, we

define the ζ-Euler numbers as follows:



Zp

ζ x e xt dμ−1x  2

ζe t 1 

∞



n0

E n,ζ t n

where En,ζ are called the ζ-Euler numbers We can show that

2

ζe t 1 

1 ζ−1

e t  ζ−1 · 2

1 ζ 

2

1 ζ

∞



n0

H n



−ζ−1 t n

where Hn−ζ−1 are the Frobenius-Euler numbers By comparing the coefficients on both sides

of1.4 and 1.5, we see that

E n,ζ 2

1 ζ H n



−ζ−1

Now, we also define the ζ-Euler polynomials as follows:

2

ζe t 1e xt

∞



n0

E n,ζx t n

In the viewpoint of1.5, we can show that

2

ζe t 1e xt  e xt

1 ζ−1

e t  ζ−1· 2

1 ζ 

2

1 ζ

∞



n0

H n



−ζ−1, x  t n

where Hn−ζ−1, x are the nth Frobenius-Euler polynomials From 1.7 and 1.8, we note that

E n,ζx  2

1 ζ H n



−ζ−1, x

1.9

cf 1 8,11–18 For each positive integer k, let Tk,ζn n

0−1

ζ   k Then we have

∞



k0

T k,ζn t k

k! ∞

k0

n



0

−1  k ζ  t k

k! n

0

−1 ζ  e t 1 −1n1 e n1t

ζe t 1 . 1.10

The ζ-Euler polynomials of order k, denoted E k n,ζ x, are defined as

e xt

2

ζe t 1

k



2

ζe t 1

× · · · ×

2

ζe t 1

e xt∞

n0

E k n,ζ x t n

Then the values of E k n,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 onZp From the properties of symmetry for the multivariate p-adic fermionic integral

onZp, 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

Let w1, w2∈ N with w1≡ 1mod 2 and w2 ≡ 1mod 2 Then we set

R mw1, w2  Zm p e w1x1x2···x m w2xt ζ w1x1···w1x m dμ−1x1 · · · dμ−1xm

Zp ζ w1w2x e w1w2xt dμ−1x

×



Zm p

e w2x1x2···x m w1yt ζ w2x1···w2x m dμ−1x1 · · · dμ−1xm,

2.1

where



Zm

p

f x1, , x mdμ−1x1 · · · dμ−1xm 



Zp

· · ·



Zp

f x1, , x mdμ−1x1 · · · dμ−1xm 2.2

Thus, we note that this expression for R m w1, w2 is symmetry in w1and w2 From2.1, we have

R m w1, w2 



Zm p

e w1x1···x m t ζ w1x1···w1x m dμ−1x1 · · · dμ−1xm e w1w2xt

×

⎛

⎝ Zp e w2x m t ζ w2x m dμ−1xm

Zp e w1w2xt ζ w1w2x dμ−1x

⎞

⎠

×



Zm−1 e w2x1···x m−1 t ζ w2x1···w2x m−1 dμ−1x1 · · · dμ−1xm−1 e w1w2yt

2.3

We can show that

Zp e xt ζ x dμ−1x

Zp e w1xt ζ w1x dμ−1x 

w1 −1

0

−1 ζ  e t∞

k0



T k,ζw1− 1 t k

By1.4 and 1.11, we see that



Zm p

e w1x1···x m t ζ w1x1···w1x m dμ−1x1 · · · dμ−1xm e w1w2xt



2

ζ w1e w1t 1

m

e w1w2xt

∞

n0

E m n,ζ w1 w2x w n

1

t n

n! .

2.5

Thus, we have

E m n,ζ w1 w2x n

0

n

 E

m

,ζ w1 w n−

From2.3, 2.4, and 2.5, we can derive

R m w1, w2 

∞



0

E m ,ζ w1 w2x w 

1

t 

!

∞



k0

T k,ζ w2 w1− 1w k2

k! t

k ∞

i0

E m−1 i,ζ w2



w1y w i

2

i! t

i



∞



0

E m ,ζ w1 w2x w 

1

t 

! ⎝∞

j0

⎛

⎝j

k0

T k,ζ w2 w1− 1w k

2w j−k2 E

m−1

j−k



w1y

k!

j − k

! j!

⎞

⎠t j

j!

⎞

⎠



∞



0

E m ,ζ w1 w2x w 

1

t 

! ⎝∞

j0

j



k0

T k,ζ w2 w1− 1

j

k E

m−1

j−k,ζ w2



w1y

w2j

⎞

⎠t j

j!

∞

n0

⎛

⎝n

j0

j



k0

T k,ζ w2 w1− 1

j

k E

m−1

j−k,ζ w2



w1y w2j w n−j1



n − j

!j! E

m

n−j,ζ w1 w2x n!

⎞

⎠ t n

n!

∞

n0

⎛

⎝n

j0

n

j w

j

2w n−j1 E m n−j,ζ w1 w2x

j



k0

T k,ζ w2 w1− 1

j

k E

m−1

j−k,ζ w2



w1y⎞⎠ t n

n! .

2.7

By the same method, we also see that

R m w1, w2 



Zm p

e w2x1···x m t ζ w2x1···w2x m dμ−1x1 · · · dμ−1xm e w1w2xt

×

⎛

⎝ Zp e w1x m t ζ w1x m dμ−1xm

Zp e w1w2xt ζ w1w2x dμ−1x

⎞

⎠

×



Zm−1

p

e w1x1···x m−1 t ζ w1x1···w1x m−1 dμ−1x1 · · · dμ−1xm−1 e w1w2yt



∞



0

E m ,ζ w2 w1x w 

2

t 

!

∞



k0

T k,ζ w1 w2− 1w k

1

t k

k!

∞



i0

E m−1 i,ζ w1



w2y

w1i t

i

i!



∞



0

E m ,ζ w2 w1x w 

2

t 

! ⎝∞

j0

⎛

⎝j

k0

T k,ζ w1 w2− 1

k!

E m−1 j−k 

w2y



j − k

!

⎞

⎠w j

1t j

⎞

⎠



∞



0

E m ,ζ w2 w1x w 

2

t 

! ⎝∞

j0

⎛

⎝j

k0

T k,ζ w1 w2− 1E m−1 j−k w2y

k!

j − k

⎞

⎠w j1t j

j!

⎞

⎠



∞

0

E m ,ζ w2 w1x w 

2

t 

! ⎝∞

j0

j

k0

j

k T k,ζ w1 w2− 1E m−1 j−k,ζ w1



w2y

w1j t

j

j!

⎞

⎠

∞

n0

⎛

⎝n

j0

j

k0

j

k T k,ζ w1 w2− 1E m−1 j−k,ζ w1



w2y w1j w n−j2 j!

n − j

!E m n−j,ζ w2 w1x n!

⎞

⎠ t n

n!

∞

n0

⎛

⎝n

j0

n

j w

j

1w n−j2 E m n−j,ζ w2 w1x

j



k0

j

k T k,ζ w1 w2− 1E m−1 j−k,ζ w1



w2y⎞⎠ t n

n! .

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



j0

n

j w

j

2w1n−j E m n−j,ζ w1 w2x

j



k0

T k,ζ w2 w1− 1

j

k E

m−1

j−k,ζ w2



w1y

n

j0

n

j w

j

1w n−j2 E n−j,ζ m w2 w1x

j



k0

j

k T k,ζ w1 w2− 1E m−1 j−k,ζ w1



w2y

.

2.9

Let y  0 and m  1 in 2.9 Then we have

n



j0

n

j w

n−j

1 w j2E n−j,ζ w1 w2x Tk,ζ w2 w1− 1

n

j0

n

j w

j

1w n−j2 E n−j,ζ w2 w1x Tk,ζ w1 w2− 1.

2.10

From2.10, we note that

n



i0

n

i w

i

1w2n−i E i,ζ w1 w2x Tn−i,ζ w2 w1− 1

n

i0

n

i w

n−i

1 w2i E i,ζ w2 w1x Tn−i,ζw1w2− 1.

2.11

If we take w2 1 in 2.11, then we have

E n,ζ w1x n

i0

n

i w

i

1E i,ζ w1 xTn−i,ζw1− 1. 2.12

From2.3, we note that

R m w1, w2 



Zm p

e w1x1···x m t ζ w1x1···w1x m dμ−1x1 · · · dμ−1xm e w1w2xt

×

⎛

⎝ Zp e w2x m t ζ w2x m dμ−1xm

Zp e w1w2xt ζ w1w2x dμ−1x

⎞

⎠

×



Zm−1 p

e w2x1···x m−1 t ζ w2x1···w2x m−1 dμ−1x1 · · · dμ−1xm−1 e w1w2yt





Zm p

e w1x1···x m t ζ w1x1···w1x m dμ−1x1 · · · dμ−1xm e w1w2xt

×

w

1 −1



i0

−1i e w2it ζ w2i

× 

Zm−1 e w2x1···x m−1 t ζ w2x1···w2x m−1 dμ−1x1 · · · dμ−1xm−1 e w1w2yt

w

1 −1



i0

−1i

ζ w2i



Zm p

e w1x1···x m w2/w1iw2xt ζ w1x1···w1x m dμ−1x1 · · · dμ−1xm

×



Zm−1 p

e w2x1···x m−1 w1yt ζ w2x1···w2x m−1 dμ−1x1 · · · dμ−1xm−1



w

1 −1



i0

−1i ζ w2i∞

k0

E m k,ζ w1

w2

w1i  w2x

w1k t

k

k!

∞



0

E ,ζ m−1 w2



w1y

w 2t



!

∞

n0

n

k0

w

1 −1



i0

−1i ζ w2i E k,ζ m w1

w2x  w2

w1i w

k

1

k! E

m−1

n−k,ζ w2



w1y  w n−k

2

n − k! n!

t n

n!

∞

n0

n

k0

n

k w

k

1w n−k2 E n−k,ζ m−1 w2



w1yw1 −1

i0

−1i

ζ w2i E m k,ζ w1

w2x  w2

w1i t

n

n! .

2.13

By the symmetric property of R m w1, w2 in w1, w2, we also see that

R m w1, w2 



Zm p

e w2x1···x m t ζ w2x1···x mdμ−1x1 · · · dμ−1xm e w1w2xt

×

⎛

⎝ Zp e w1x m t ζ w1x m dμ−1xm

Zp e w1w2xt ζ w1w2x dμ−1x

⎞

⎠

×



Zm−1 p

e w1x1···x m−1 t ζ w1x1···x m−1dμ−1x1 · · · dμ−1xm−1 e w1w2yt





Zm p

e w2x1···x m t ζ w2x1···x mdμ−1x1 · · · dμ−1xm e w1w2xt

×

w

2 −1



i0

−1i

e w1it ζ w1i

×



Zm−1 p

e w1x1···x m−1 w2yt ζ w1x1···x m−1dμ−1x1 · · · dμ−1xm−1

w2−1

i0

−1i ζ w1i



Zm p

e w2x1···x m w1/w2iw1xt ζ w2x1···x mdμ−1x1 · · · dμ−1xm

×



Zm−1 p

e w1x1···x m−1 w2yt ζ w1x1···w1x m−1 dμ−1x1 · · · dμ−1xm−1



w

2 −1



i0

−1i ζ w1i∞

k0

E m k,ζ w2

w1

w2i  w1x

w k

2

t k

k!

∞



0

E m−1 ,ζ w1 

w2y

w 

1

t 

!

n0

n



k0

w

2 −1



i0

−1i

ζ w1i E m k,ζ w2

w1x  w1

w2i w

k

2

k! E

m−1

n−k



w2y  w n−k

1

n − k! n!

t n

n!

∞

n0

n

k0

n

k w

k

2w1n−k E m−1 n−k,ζ w1



w2yw2 −1

i0

−1i ζ w1i E m k,ζ w2

w1x  w1

w2i t

n

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



k0

n

k w

k

1w2n−k E m−1 n−k,ζ w2



w1yw1 −1

i0

−1i ζ w2i E m k,ζ w1

w2x  w2

w1i

n

k0

n

k w

k

2w n−k1 E m−1 n−k,ζ w1

w2yw2 −1

i0

−1i ζ w1i E k,ζ m w2

w1x  w1

w2i

.

2.15

Let y  0 and m  1, we have

w1n

w1 −1

i0

−1i ζ w2i E n,ζ w1

w2x  w2

w1i

 w n

2

w2 −1

i0

−1i ζ w1i E n,ζ w2

w1x  w1

w2i

From2.16, we can derive

w1 −1

i0

−1i ζ i E n,ζ w1

x  1

w1i

 1

w n

1

E n,ζw1x . 2.17

Acknowledgment

The present research has been conducted by the research grant of the Kwangwoon University

in 2009

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onZp, 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,... purpose of this paper is to investigate some properties of symmetry for the multivariate p-adic fermionic

integral onZp From the properties of symmetry for the multivariate p-adic fermionic...

Then the values of E k n,ζ x at x  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