On the Identities of Symmetry for the ζ-Euler Polynomials of Higher Order

doi:10.1155/2009/273545

Research Article

On the Identities of Symmetry for the ζ-Euler Polynomials of Higher Order

Taekyun Kim,

Kyoung Ho Park,

and Kyung-won Hwang

1Division 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 1q^pN

p^N−1 x0

fx

−q_x

1.1

see1. Let us define the fermionicp-adic invariant integral onZpas follows:

I−1 f

lim

q→1I−q f

Zp

fxdμ−1x 1.2

see1–8. From1.2, we have I−1

f1

I−1 f

2f0 1.3

see9,10, wheref1x fx1. Forζ∈Cpwith|1−ζ|_p <1, letfx e^xtζ^x. Then, we define theζ-Euler numbers as follows:

Zp

ζ^xe^xtdμ₋₁x 2

ζe^t1 ^∞

n0

En,ζtⁿ

n!, 1.4

whereEn,ζare called theζ-Euler numbers. We can show that 2

ζe^t1 1ζ⁻¹ e^tζ⁻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

ζe^t1e^xt∞

n0

En,ζxtⁿ

n!. 1.7

In the viewpoint of1.5, we can show that 2

ζe^t1e^xte^xt1ζ⁻¹ e^tζ⁻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 _n

0−1ζ^k. Then we have ∞

k0

Tk,ζnt^k k! ^∞

k0

n 0

−1^kζ t^k

k! ⁿ

0

−1ζe^t 1 −1ⁿ¹e^n1t

ζe^t1 . 1.10

Theζ-Euler polynomials of orderk, denotedE^k_n,ζx, are defined as

e^xt 2

ζe^t1 _k

2

ζe^t1

× · · · × 2

ζe^t1

e^xt∞

n0

E^k_n,ζxtⁿ

n!. 1.11

Then the values ofE^k_n,ζxatx 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

R^mw1, w2

Z^mpe^{w1x1x2···xmw2xt}ζ^{w1x1···w1xm}dμ−1x1· · ·dμ−1xm

Zpζ^w1w2xe^w1w2xtdμ−1x

×

Z^mp

e^{w2x1x2···xmw1yt}ζ^{w2x1···w2xm}dμ−1x1· · ·dμ−1xm,

2.1

where

Z^mp

fx1, . . . , xmdμ₋₁x1· · ·dμ−1xm

Zp

· · ·

Zp

fx1, . . . , xmdμ₋₁x1· · ·dμ−1xm. 2.2

Thus, we note that this expression forR^mw1, w2is symmetry inw1andw2. From2.1, we have

R^mw1, w2

Z^mp

e^{w1x1···xmt}ζ^{w1x1···w1xm}dμ₋₁x1· · ·dμ₋₁xm

e^w1w2xt

×

⎛

⎝

Zpe^w2xmtζ^w2xmdμ₋₁xm

Zpe^w1w2xtζ^w1w2xdμ−1x

⎞

⎠

×

Z^m−1p

e^{w2x1···xm−1t}ζ^{w2x1···w2xm−1}dμ−1x1· · ·dμ−1xm−1

e^w1w2yt. 2.3

We can show that

Zpe^xtζ^xdμ−1x

Zpe^w1xtζ^w1xdμ−1x ^w1−1

0

−1ζe^t∞

k0

Tk,ζw1−1t^k

k!. 2.4

By1.4and1.11, we see that

Z^mp

e^{w1x1···xmt}ζ^{w1x1···w1xm}dμ₋₁x1· · ·dμ₋₁xm

e^w1w2xt

2 ζ^w1e^w1t1

_m e^w1w2xt

^∞

n0

E^m_n,ζw1w2xw₁ⁿtⁿ n!.

2.5

Thus, we have

E^m_n,ζw1w2x ⁿ

0

n

E^m_,ζw1w₂ⁿ⁻xⁿ⁻. 2.6

From2.3,2.4, and2.5, we can derive

R^mw1, w2

∞ 0

E^m_,ζw1w2xw₁t

! ∞ k0

Tk,ζ^w2w1−1w^k₂ k! t^k

∞ i0

E^m−1_i,ζw2

w1yw₂ⁱ i! tⁱ

^∞

0

E^m_,ζw1w2xw₁t

!

⎛

⎝^∞

j0

⎛

⎝^j

k0

Tk,ζ^w2w1−1w^k₂w^j−k₂ E^m−1_j−k w1y k!

j−k

! j!

⎞

⎠t^j j!

⎞

⎠

^∞

0

E^m_,ζw1w2xw₁t

!

⎛

⎝^∞

j0

j k0

Tk,ζ^w2w1−1 j k

E^m−1_j−k,ζw2

w1y w₂^j

⎞

⎠t^j j!

^∞

n0

⎛

⎝ⁿ

j0

j k0

Tk,ζ^w2w1−1 j k

E^m−1_j−k,ζw2

w1y

w₂^jw^n−j₁ n−j

!j!E^m_n−j,ζw1w2xn!

⎞

⎠tⁿ n!

^∞

n0

⎛

⎝ⁿ

j0

n j

w^j₂w^n−j₁ E^m_n−j,ζw1w2x j k0

Tk,ζ^w2w1−1 j k

E^m−1_j−k,ζw2

w1y⎞

⎠tⁿ n!.

2.7

By the same method, we also see that

R^mw1, w2

Z^mp

e^{w2x1···xmt}ζ^{w2x1···w2xm}dμ−1x1· · ·dμ−1xm

e^w1w2xt

×

⎛

⎝

Zpe^w1xmtζ^w1xmdμ−1xm

Zpe^w1w2xtζ^w1w2xdμ₋₁x

⎞

⎠

× Z^m−1p

e^{w1x1···xm−1t}ζ^{w1x1···w1xm−1}dμ₋₁x1· · ·dμ₋₁x_m−1

e^w1w2yt

^∞

0

E^m_,ζw2w1xw₂t

! ∞ k0

Tk,ζ^w1w2−1w^k₁t^k k!

∞ i0

E^m−1_i,ζw1

w2y w₁ⁱtⁱ

i!

^∞

0

E^m_,ζw2w1xw₂t

!

⎛

⎝^∞

j0

⎛

⎝^j

k0

Tk,ζ^w1w2−1 k!

E^m−1_j−k w2y j−k

!

⎞

⎠w^j₁t^j

⎞

⎠

^∞

0

E^m_,ζw2w1xw₂t

!

⎛

⎝^∞

j0

⎛

⎝^j

k0

Tk,ζ^w1w2−1E^m−1_j−k w2y k!

j−k

! j!

⎞

⎠w^j₁t^j j!

⎞

⎠

^∞

0

E^m_,ζw2w1xw₂t

!

⎛

⎝^∞

j0

j k0

j k

Tk,ζ^w1w2−1E^m−1_j−k,ζw1

w2y w₁^jt^j

j!

⎞

⎠

^∞

n0

⎛

⎝ⁿ

j0

j k0

j k

Tk,ζ^w1w2−1E^m−1_j−k,ζw1

w2y

w₁^jw^n−j₂ j!

n−j

!E^m_n−j,ζw2w1xn!

⎞

⎠tⁿ n!

^∞

n0

⎛

⎝ⁿ

j0

n j

w^j₁w^n−j₂ E^m_n−j,ζw2w1x j k0

j k

Tk,ζ^w1w2−1E^m−1_j−k,ζw1

w2y⎞

⎠tⁿ n!.

2.8

By comparing the coefficients on both sides of2.7and2.8, we obtain the following.

Theorem 2.1. Forw1, w2∈Nwithw1≡1mod 2,w2≡1mod 2, andn≥0, m≥1, one has

n j0

n j

w₂^jw₁^n−jE^m_n−j,ζw1w2x j k0

Tk,ζ^w2w1−1 j k

E^m−1_j−k,ζw2

w1y

ⁿ

j0

n j

w₁^jw^n−j₂ E_n−j,ζ^m w2w1x j k0

j k

Tk,ζ^w1w2−1E^m−1_j−k,ζw1

w2y .

2.9

Lety0 andm1 in2.9. Then we have

n j0

n j

w^n−j₁ w^j₂E_n−j,ζ^w1w2xTk,ζ^w2w1−1

ⁿ

j0

n j

w^j₁w^n−j₂ E_n−j,ζ^w2w1xTk,ζ^w1w2−1.

2.10

From2.10, we note that

n i0

n i

wⁱ₁w₂ⁿ⁻ⁱEi,ζ^w1w2xTn−i,ζ^w2w1−1

ⁿ

i0

n i

wⁿ⁻ⁱ₁ w₂ⁱEi,ζ^w2w1xTn−i,ζ^w1w2−1.

2.11

If we takew21 in2.11, then we have

En,ζw1x ⁿ

i0

n i

w₁ⁱEi,ζ^w1xTn−i,ζw1−1. 2.12

From2.3, we note that

R^mw1, w2

Z^mp

e^{w1x1···xmt}ζ^{w1x1···w1xm}dμ−1x1· · ·dμ−1xm

e^w1w2xt

×

⎛

⎝

Zpe^w2xmtζ^w2xmdμ−1xm

Zpe^w1w2xtζ^w1w2xdμ−1x

⎞

⎠

× Z^m−1p

e^{w2x1···xm−1t}ζ^{w2x1···w2xm−1}dμ−1x1· · ·dμ−1xm−1

e^w1w2yt

Z^mp

e^{w1x1···xmt}ζ^{w1x1···w1xm}dμ₋₁x1· · ·dμ₋₁xm

e^w1w2xt

× ^w1−1

i0

−1ⁱe^w2itζ^w2i

×

Z^m−1p

e^{w2x1···xm−1t}ζ^{w2x1···w2xm−1}dμ−1x1· · ·dμ−1xm−1

e^w1w2yt

^w1−1

i0

−1ⁱζ^w2i

Z^mp

e^{w1x1···xmw2/w1iw2xt}ζ^{w1x1···w1xm}dμ−1x1· · ·dμ−1xm

× Z^m−1p

e^{w2x1···xm−1w1yt}ζ^{w2x1···w2xm−1}dμ−1x1· · ·dμ−1xm−1

^w1−1

i0

−1ⁱζ^w2i ∞ k0

E^m_k,ζw1

w2

w1iw2x

w₁^kt^k k!

∞ 0

E_,ζ^m−1w2

w1y w₂t

!

^∞

n0

n k0

w1−1 i0

−1ⁱζ^w2iE_k,ζ^mw1

w2xw2

w1i w^k₁

k!E^m−1_n−k,ζw2

w1y w^n−k₂ n−k!n!

tⁿ n!

^∞

n0

n k0

n k

w₁^kw^n−k₂ E_n−k,ζ^m−1w2

w1y^w1−1

i0

−1ⁱζ^w2iE^m_k,ζw1

w2xw2

w1i tⁿ

n!. 2.13

By the symmetric property ofR^mw1, w2inw1, w2, we also see that

R^mw1, w2

Z^mp

e^{w2x1···xmt}ζ^w2x1···xmdμ−1x1· · ·dμ−1xm

e^w1w2xt

×

⎛

⎝

Zpe^w1xmtζ^w1xmdμ₋₁xm

Zpe^w1w2xtζ^w1w2xdμ₋₁x

⎞

⎠

× Z^m−1p

e^{w1x1···xm−1t}ζ^{w1x1···xm−1}dμ₋₁x1· · ·dμ₋₁x_m−1

e^w1w2yt

Z^mp

e^{w2x1···xmt}ζ^w2x1···xmdμ−1x1· · ·dμ−1xm

e^w1w2xt

× ^w2−1

i0

−1ⁱe^w1itζ^w1i

× Z^m−1p

e^{w1x1···xm−1w2yt}ζ^{w1x1···xm−1}dμ−1x1· · ·dμ−1xm−1

^w2−1

i0

−1ⁱζ^w1i

Z^mp

e^{w2x1···xmw1/w2iw1xt}ζ^w2x1···xmdμ−1x1· · ·dμ−1xm

× Z^m−1p

e^{w1x1···xm−1w2yt}ζ^{w1x1···w1xm−1}dμ₋₁x1· · ·dμ₋₁x_m−1

^w2−1

i0

−1ⁱζ^w1i ∞ k0

E^m_k,ζw2

w1

w2iw1x

w^k₂t^k k!

∞ 0

E^m−1_,ζw1

w2y w₁t

!

^∞

n0

n k0

w2−1 i0

−1ⁱζ^w1iE^m_k,ζw2

w1x w1

w2i w₂^k

k!E^m−1_n−k

w2y w^n−k₁ n−k!n!

tⁿ n!

^∞

n0

n k0

n k

w^k₂w₁^n−kE^m−1_n−k,ζw1

w2y^w2−1

i0

−1ⁱζ^w1iE^m_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

w^k₁w₂^n−kE^m−1_n−k,ζw2

w1y^w1−1

i0

−1ⁱζ^w2iE^m_k,ζw1

w2xw2

w1i

ⁿ

k0

n k

w^k₂w^n−k₁ E^m−1_n−k,ζw1

w2y^w2−1

i0

−1ⁱζ^w1iE_k,ζ^mw2

w1xw1

w2i

.

2.15

Lety0 andm1, we have

w₁ⁿ

w1−1 i0

−1ⁱζ^w2iEn,ζ^w1

w2xw2

w1i

wⁿ₂

w2−1 i0

−1ⁱζ^w1iEn,ζ^w2

w1xw1

w2i

. 2.16

From2.16, we can derive

w1−1 i0

−1ⁱζⁱEn,ζ^w1

x 1

w1i

1

wⁿ₁En,ζw1x. 2.17

Acknowledgment

The present research has been conducted by the research grant of the Kwangwoon University in 2009.

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