On the Symmetric Properties for the Generalized Twisted Bernoulli Polynomials

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Volume 2009, Article ID 164743,8pages doi:10.1155/2009/164743

Research Article

On the Symmetric Properties for the Generalized

Twisted Bernoulli Polynomials

Taekyun Kim and Young-Hee Kim

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

Correspondence should be addressed to Young-Hee Kim,[email protected]

Received 6 July 2009; Accepted 18 October 2009

Recommended by Narendra Kumar Govil

We study the symmetry for the generalized twisted Bernoulli polynomials and numbers. We give some interesting identities of the power sums and the generalized twisted Bernoulli polynomials using the symmetric properties for thep-adic invariant integral.

Copyrightq2009 T. Kim and Y.-H. Kim. 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

Letpbe a fixed prime number. Throughout this paper, the symbolsZ,Zp,Qp, andCpdenote

the ring of rational integers, the ring ofp-adic integers, the field ofp-adic rational numbers, and the completion of algebraic closure ofQp, respectively. LetNbe the set of natural numbers

andZN∪{0}. Letνpbe the normalized exponential valuation ofCpwith|p|_pp−νppp−1.

LetUDZpbe the space of uniformly diﬀerentiable function onZp. Forf∈UDZp,

thep-adic invariant integral onZpis defined as

If

ZpfxdxNlim→ ∞

1

pN pN₋₁

x0

fx 1.1

see1. From1.1, we note that

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wheref0 dfx/dx|x0andf1x fx1. Forn∈N, letfnx fxn. Then we can

derive the following equation from1.2:

Ifn

If

n−1

i0

fi 1.3

see1–7.

Letdbe a fixed positive integer. Forn∈N, let

XXdlim←−_NZ/dpNZ, X1Zp,

X∗ 0<a<dp

a,p1

adpZp

,

adpNZp

x∈X|x≡amoddpN ,

1.4

wherea∈Zlies in 0≤a < dpN_{. It is easy to see that}

X

fxdx

Zpfxdx, forf∈UD

Zp

. 1.5

The ordinary Bernoulli polynomialsBnxare defined as

t et₋₁e

xt∞

n0 Bnxt

n

n!, 1.6

and the Bernoulli numbersBnare defined asBnBn0 see1–19.

Forn∈N, letTpbe thep-adic locally constant space defined by

Tp

n≥1

Cpn lim

n→ ∞Cpn, 1.7

whereCpn {ω | ωp n

1}is the cyclic group of orderpn_{. It is well known that the twisted}

Bernoulli polynomials are defined as

t ξet₋₁e

xt∞

n0

Bn,ξxt n

n!, ξ∈Tp, 1.8

and the twisted Bernoulli numbersBn,ξare defined asBn,ξ Bn,ξ0 see15–18.

Let χ be Dirichlet’s character with conductor d ∈ N. Then the generalized twisted

Bernoulli polynomialsBn,χ,ξxattached toχare defined as follows:

d−1

a0

χaξa_eat_t

ξd_edt₋₁ e xt∞

n0

Bn,χ,ξxt n

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The generalized twisted Bernoulli numbers attached to χ, Bn,χ,ξ, are defined as Bn,χ,ξ

Bn,χ,ξ0 see16.

Recently, many authors have studied the symmetric properties of thep-adic invariant integrals on Zp, which gave some interesting identities for the Bernoulli and the Euler

polynomialscf.3,6,7,13,14,20–27. The authors of this paper have established various

identities by the symmetric properties of the p-adic invariant integrals and investigated interesting relationships between the power sums and the Bernoulli polynomials see 2,3,6,7,13.

The twisted Bernoulli polynomials and numbers and the twisted Euler polynomials and numbers are very important in several fields of mathematics and physicscf. 15–18.

The second author has been interested in the twisted Euler numbers and polynomials and the twisted Bernoulli polynomials and studied the symmetry of power sum and twisted Bernoulli polynomialssee11–13.

The purpose of this paper is to study the symmetry for the generalized twisted Bernoulli polynomials and numbers attached toχ. InSection 2, we give interesting identities for the power sums and the generalized twisted Bernoulli polynomials using the symmetric properties for thep-adic invariant integral.

2. Symmetry for the Generalized Twisted Bernoulli Polynomials

Letχbe Dirichlet’s character with conductord∈N. Forξ∈Tp, we have

X

χxξxextdx t

_d₋₁

i0 χiξieit ξd_edt₋₁

∞

n0 Bn,χ,ξt

n

n!, 2.1

whereBn,χ,ξare thenth generalized twisted Bernoulli numbers attached toχ. We also see that

the generalized twisted Bernoulli polynomials attached toχare given by

X

χyξyexytdy t

d−1

i0 χiξieit ξd_edt₋₁ e

xt∞

n0

Bn,χ,ξxt n

n!. 2.2

By2.1and2.2, we see that

X

χxξx_xn_dx_B n,χ,ξ,

X

χyξy_x_yn

dyBn,χ,ξx. 2.3

From2.3, we derive that

Bn,χ,ξx n

l0

n l

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By1.5and2.3, we see that

X

χxξxextdx 1 d

d−1

a0

χaeatξa

Zpξ

dx_edxt_dx 1

d

d−1

a0

χaξa dt ξd_edt₋₁e

a/ddt_. _2.5

From2.2and2.5, we obtain that

X

χxξxextdx

∞

n0

dn−1

d−1

a0

χaξaBn,ξd _a

d tn

n!. 2.6

Thus we have the following theorem from2.1and2.6.

Theorem 2.1. Forξ∈Tp, one has

Bn,χ,ξ dn−1 d−1

a0

χaξaBn,ξd _a

d . 2.7

By1.3and1.5, we have that forn∈N,

X

fxndx

X

fxdx

n−1

i0

fi, 2.8

wherefi dfx/dx|xi. Takingfx χxξxextin2.8, it follows that

1

t

X

χxξndxendxtdx−

X

χxξxextdx

nd

Xχxξxextdx

Xξndxendxtdx

ξndendt−1

ξd_edt₋₁

_d₋₁

i0

χiξieit

.

2.9

Thus, we have

1

t

X

χxξndxendxtdx−

X

χxξxextdx

∞

k0

_nd₋₁

l0

χlξllk

tk

k!. 2.10

Fork∈Z, let us define thep-adic functionalKχ, ξ, k:nas follows:

Kχ, ξ, k:n

n

l0

χlξllk. 2.11

By2.10and2.11, we see that fork, n, d∈N,

X

χxξndxndxkdx−

X

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From2.3and2.12, we have the following result.

Theorem 2.2. Forξ∈Tpandk, n, d∈N, one has

ξndBk,χ,ξnd−Bk,χ,ξkK

χ, ξ, k−1 :nd−1. 2.13

Letw1, w2, d∈N. Then we have that

d_X_Xχx1χx2ξw1x1w2x2_ew1x1w2x2t_dx1dx2

Xξdw1w2xedw1w2xtdx

t

ξdw1w2_edw1w2t−₁

ξw1dedw1t−1ξw2dedw2t−1 _d₋₁

a0

χaξw1a_ew1at _d₋₁

b0

χbξw2b_ew2bt

.

2.14

By2.9,2.10, and2.11, we see that

w1d_Xχxξx_ext_dx

Xξdw1xedw1xtdx

∞

k0

Kχ, ξ, k:dw1−1t

k

k!. 2.15

Now let us define thep-adic functionalYχ,ξw1, w2as follows:

Yχ,ξw1, w2

d_X_Xχx1χx2ξw1x1w2x2_ew1x1w2x2w1w2xt_dx1dx2

Xξdw1w2x3edw1w2x3tdx3

. 2.16

Then it follows from2.14that

Yχ,ξw1, w2

tξdw1w2_edw1w2t−₁_ew1w2xt

ξw1dedw1t−1ξw2dedw2t−1 _d₋₁

a0

χaξw1a_ew1at _d₋₁

b0

χbξw2b_ew2bt

.

2.17

By2.15and2.16, we obtain that

Yχ,ξw1, w2

1

w1

X

χx1ξw1x1_ew1x1w2xt_dx1

_dw1

Xχx2ξw2x2ew2x2tdx2

∞

l0

_l

i0

l i

Bi,χ,ξw1w2xK

χ, ξw2_{, l}−_i_:_dw1−₁_wi−1

1 w2l−i

tl

l!.

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On the other hand, the symmetric property ofYχ,ξw1, w2shows that

Yχ,ξw1, w2

1

w2

X

χx2ξw2x2_ew2x2w1xt_dx2

_dw2

∞

l0

_l

i0

l i

Bi,χ,ξw₂w1xKχ, ξw1, l−i:dw2−1wi−1

2 wl1−i

tl

l!.

2.19

Comparing the coeﬃcients on the both sides of 2.18 and 2.19, we have the following

theorem.

Theorem 2.3. Letξ∈Tpandd, w1, w2∈N. Then one has

l

i0

l i

Bi,χ,ξw1w2xK

χ, ξw2_{, l}−_i_:_dw1−₁_wi−1

1 wl2−i

l

i0

l i

Bi,χ,ξw2w1xK

χ, ξw1_{, l}−_i_:_dw2−₁_wi−1

2 wl1−i.

2.20

We also derive some identities for the generalized twisted Bernoulli numbers. Taking

x0 inTheorem 2.3, we have the following corollary.

Corollary 2.4. Letξ∈Tpandd, w1, w2∈N. Then one has

l

i0

l i

Bi,χ,ξw1K

χ, ξw2_{, l}−_i_:_dw1−₁_wi−1

1 wl2−i

l

i0

l i

Bi,χ,ξw2K

χ, ξw1_{, l}−_i_:_dw2−₁_wi−1

2 wl1−i.

2.21

Now we will derive another identities for the generalized twisted Bernoulli polynomials using the symmetric property ofYχ,ξw1, w2. From1.2,2.15and2.17, we

see that

Yχ,ξw1, w2

ew1w2xt

w1

X

χx1ξw1x1_ew1x1t_dx1

_dw1

1

w1

dw1−1

i0

χiξw2i

X

χx1ξw1x1_ew1x1w2xw2/w1it_dx1

∞

k0

_dw 1−1

i0

χiξw2i_B k,χ,ξw1

w2xw2 w1i

wk−1 1

tk

k!.

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From the symmetric property ofYχ,ξw1, w2, we also see that

Yχ,ξw1, w2

ew1w2xt

w2

X

χx2ξw2x2_ew2x2t_dx2

_dw2

1

w2

dw2−1

i0

χiξw1i

X

χx2ξw2x2_ew2x2w1xw1/w2it_dx2

∞

k0

_dw 2−1

i0

w1xw1 w2i

wk−1 2

tk

k!.

2.23

Comparing the coeﬃcients on the both sides of2.22and 2.23, we obtain the following

theorem.

Theorem 2.5. Letξ∈Tpandd, w1, w2∈N. Then one has

dw1−1

i0

w2xw2 w1i

w₁k−1

dw2−1

i0

w1xw1 w2i

wk₂−1. 2.24

If we take x 0 in Theorem 2.5, we also derive the interesting identity for the generalized twisted Bernoulli numbers as follows: ford, w1, w2∈N,

dw1−1

i0

χiξw2i_B k,χ,ξw₁

w2 w1i

wk₁−1

dw2−1

i0

χiξw1i_B k,χ,ξw₂

w1 w2i

wk₂−1. 2.25

Acknowledgment

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

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