Symmetry Properties of Higher-Order Bernoulli Polynomials

doi:10.1155/2009/318639

Research Article

Symmetry Properties of Higher-Order Bernoulli Polynomials

Taekyun Kim,

Kyung-Won Hwang,

and Young-Hee Kim

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

2Department of General Education, Kookmin University, Seoul 136-702, South Korea

Correspondence should be addressed to Taekyun Kim,[email protected] Kyung-Won Hwang,[email protected]

Received 11 March 2009; Revised 6 July 2009; Accepted 2 August 2009 Recommended by Patricia J. Y. Wong

We investigate properties of identities and some interesting identities of symmetry for the Bernoulli polynomials of higher order using the multivariatep-adic invariant integral onZp. 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

Let p be a fixed prime number. Throughout this paper Zp,Qp, and Cp will, respectively, denote the ring of p-adic rational integers, the field of p-adic rational numbers, and the completion of algebraic closure ofQp. Forx∈Cp, we use the notationx_q 1−q^x/1−q.

Let UDZp be the space of uniformly differentiable functions on Zp, and let vp be the normalized exponential valuation ofCpwith|p|pp^−vpp1/p. Forq∈Cpwith|1−q|p<1, theq-Volkenborn integral onZpis defined as

Iq

f

Zp

fxdμqx lim

N→ ∞

1 p^N

q p^N−1

x0

fxq^x, f ∈UD Zp

1.1

see1,2. The ordinaryp-adic invariant integral onZpis given by

I1

f lim

q→1Iq

f

Zp

fxdx 1.2

see1–15. Letf0 dfx/dx|x0. Then we easily see that I1

f1

I1

f

f0, wheref1x fx1. 1.3

From1.3, we can derive

Zp

e^xtdx t

e^t−1 ^∞

n0

Bntⁿ

n! 1.4

see2,8–10, whereBnare thenth Bernoulli numbers.

By1.2and1.3, we easily see that n

Zpe^xtdx

Zpe^nxtdx 1

t _Zpe^xntdx−

Zp

e^xtdx

ⁿ⁻¹

i0

e^it∞

n0 n−1

i0

i^k t^k

k! ^∞

k0

Skn−1t^k k!,

1.5

whereSkn 0^k1^k· · ·n^kfork∈Z.

It is known that the Bernoulli polynomials are defined by

Zp

e^xytdx t

e^t−1e^xt∞

n0

Bnxtⁿ

n!, 1.6

whereBnxare called thenth Bernoulli polynomials. The Bernoulli polynomials of orderk, denotedB^k_nx, are defined as

e^xt t

e^t−1 _k

t

e^t−1

× · · · × t

e^t−1

e^xt∞

n0

B^kn xtⁿ

n! 1.7

see3–6. Then the values of Bn^kxat x 0 are called the Bernoulli numbers of order k. When k 1, the polynomials or numbers are called the Bernoulli polynomials or numbers. The purpose of this paper is to investigate some interesting properties of symmetry for the multivariate p-adic invariant integral on Zp. From the properties of symmetry for the multivariate p-adic invariant integral on Zp, we derive some interesting identities of symmetry for the Bernoulli polynomials of higher order.

2. Symmetry Properties of Higher-Order Bernoulli Polynomials

Letw1, w2∈N. Then we define

D^mw1, w2

w1t

− _m

e^w1w2tx

e^w1w2t−1 w2t

− _m

e^w1w2yt

. 2.1

From2.1, we note that

D^mw1, w2

Z^mpe^{w1x1x2···xmw2xt}dx1· · ·dxm

Z^mpe^{w2x1x2···xmw1yt}dx1· · ·dxm

Zpe^w1w2xtdx , 2.2

where

Z^mpfx1, . . . , xmdx1· · ·dxm

Zp· · ·

Zpfx1, . . . , xmdx1· · ·dxm.

In2.1, we note thatD^mw1, w2is symmetric inw1, w2. By2.1, we see that

D^mw1, w2

Z^mp

e^{w1x1···xmt}dx1· · ·dxm

e^w1w2xt

⎛

⎝

Zpe^w2xmtdxm

Zpe^w1w2xtdx

⎞

⎠

×

Z^m−1p

e^{w2x1···xm−1t}dx1· · ·dxm−1

e^w1w2yt.

2.3

It is easy to see that

e^w1w2xt

Z^mp

e^{w1x1···xmt}dx1· · ·dxm

w1t e^w1t−1

m

e^w1w2xt∞

n0

Bn^mw2xw₁ⁿtⁿ

n!. 2.4

From2.1,2.3, and the above formula, we can derive

D^mw1, w2

∞ 0

B^m w2xw₁t

! ∞ k0

Skw1−1w₂^k k!t^k

∞ i0

B^m−1_i

w1ywⁱ₂ i! tⁱ

1 w1

^∞

0

B^m w2xw₁⁻¹t

!

⎛

⎝^∞

j0

⎛

⎝^j

k0

Skw1−1w₂^kw^j−k₂ B_j−k^m−1 w1y k!

j−k

! j!

⎞

⎠t^j j!

⎞

⎠

^∞

n0

⎛

⎝ⁿ

j0

n j

w^j₂w₁^n−j−1B_n−j^mw2x j k0

Skw1−1 j k

B^m−1_j−k

w1y⎞

⎠tⁿ n!.

2.5 By the symmetry ofD^mw1, w2inw1andw2, we see that

D^mw1, w2

∞ n0

⎛

⎝ⁿ

j0

n j

w^j₁w₂^n−j−1B_n−j^mw1x j k0

j k

Skw2−1B^m−1_j−k

w2y⎞

⎠tⁿ n!.

2.6

By comparing the coefficients on both sides of 2.5 and 2.6, we obtain the following theorem.

Theorem 2.1. Forw1, w2∈N, n≥0, m≥1, one has

n j0

n j

w₂^jw^n−j−1₁ B^m_n−jw2x j k0

Skw1−1 j k

B_j−k^m−1

w1y

ⁿ

j0

n j

w₁^jw^n−j−1₂ B^m_n−jw1x j k0

j k

Skw2−1B_j−k^m−1 w2y

.

2.7

Lety0 andm1 in2.7. Then we have the following corollary.

Corollary 2.2. Forn∈Z, one has

n j0

n j

w^n−j−1₁ w^j₂B_n−jw2xSjw1−1

ⁿ

j0

n j

w^j₁w₂^n−j−1Bn−jw1xSjw2−1.

2.8

If we takew21 in2.8, then we also obtain the following corollary.

Corollary 2.3. Forw1∈N,one has

Bnw1x ⁿ

i0

n i

w₁ⁱ⁻¹BixS_n−iw1−1. 2.9

By the definition ofD^mw1, w2, we easily see that

D^mw1, w2

w1t e^w1t−1

_m

e^xw1w2te^w1w2t−1 e^w2t−1

w2t e^w2t−1

_m−1

e^yw1w2t 1 w1

1 w1

w1−1 i0

∞ k0

B^m_k

w2x w2

w1i

w^k₁t^k k!

∞ 0

B^m−1 w1y

w₂t

!

^∞

n0

n k0

w1−1 i0

B_k^m

w2xw2

w1i

w^k−1₁ k! B^m−1_n−k

w1y w^n−k₂ n−k!n!

tⁿ n!

^{∞ n} n

w^k−1₁ w^n−k₂ B_n−k^m−1

w1y^w1−1 B^m_k

w2xw2

w i tⁿ

n!.

2.10

From the symmetric property ofD^mw1, w2inw1, w2, we note that

D^mw1, w2

∞ n0

n k0

n k

w₂^k−1w₁^n−kB^m−1_n−k

w2y^w2−1

i0

B_k^m

w1xw1

w2i tⁿ

n!. 2.11

By comparing the coefficients on both sides of 2.10 and 2.11, we obtain the following theorem.

Theorem 2.4. Forw1, w2∈N, n∈Z, m∈N, one has

n k0

n k

w^k−1₁ w₂^n−kB_n−k^m−1

w1y^w1−1

i0

B_k^m

w2xw2

w1i

ⁿ

k0

n k

w₂^k−1w₁^n−kB^m−1_n−k

w2y^w2−1

i0

B_k^m

w1xw1

w2i

.

2.12

Lety0 andm1 in2.12. Then we obtain the followingCorollary 2.5.

Corollary 2.5. Forw1, w2∈N, one has w₁ⁿ⁻¹

w1−1 i0

Bn

w2xw2

w1i

wⁿ⁻¹₂

w2−1 i0

Bn

w1xw1

w2i

. 2.13

From2.12, we can get the well-known result due to Raabe:

w1−1 i0

Bn

x 1

w1i

w¹⁻ⁿ₁ Bnw1x. 2.14

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