Some Identities on Bernoulli and Euler Numbers

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Discrete Dynamics in Nature and Society Volume 2012, Article ID 486158,12pages doi:10.1155/2012/486158

Research Article

Some Identities on Bernoulli and Euler Numbers

D. S. Kim,

¹

T. Kim,

²

J. Choi,

³

and Y. H. Kim

³

1Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea

2Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea

3Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea

Correspondence should be addressed to T. Kim,[email protected] Received 15 November 2011; Accepted 23 December 2011 Academic Editor: Delfim F. M. Torres

Copyrightq2012 D. S. 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.

Recently, Kim introduced the fermionic p-adic integral on Zp. By using the equations of the fermionic and bosonic p-adic integral onZp, we give some interesting identities on Bernoulli and Euler numbers.

1. Introduction/Preliminaries

Letpbe a fixed odd prime number. Throughout this paper,Zp,Qp, andCp will denote 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}. Thep-adic absolute value| · |pis normally defined by|p|p1/p.

Let UDZpbe the space of uniformly diﬀerentiable functions onZp and CZpthe space of continuous function on Zp. For f ∈ CZp, the fermionicp-adic integral on Zp is defined by Kim as follows:

I−1 f

Zp

fxdμ−1x lim

N→ ∞ p^N−1

x0

fx−1^x, see1. 1.1

The following fermionicp-adic integral equation onZpis well knownsee1–3:

I₋₁ f1

I₋₁ f

2f0, 1.2

wheref1x fx1.

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From1.1and 1.2, we can derive the generating function of Euler polynomials as follows:

Zp

e^xytdμ−1 y

2

e^t1e^xt^∞

n0

Enxtⁿ

n!, 1.3

where Enxis the nth ordinary Euler polynomial see1–4. In the special case, x 0, En0 Enis called thenth ordinary Euler number.

By1.3, we get Witt’s formula for thenth Euler polynomial as follows:

Zp

xy_n dμ−1

y

Enx, forn∈Z. 1.4

Thus, by1.4, we have

Enx Exⁿⁿ

l0

n l

x^n−lEl, 1.5

with the usual convention about replacingEⁿbyEnsee5,6. From1.3, we note that E1ⁿEn2δ0,n, 1.6

whereδk,nis the Kronecker symbolsee3. By1.2and1.4, we get

Zp

xy1_n dμ−1

y

Zp

xy_n dμ−1

y

2xⁿ. 1.7

Thus, by1.4and1.7, we have

Enx1 Enx 2xⁿ, forn∈Z. 1.8 Equation1.8is equivalent to

xⁿEnx 1 2

n−1 l0

n l

Elx. 1.9

From1.6, we can derive the following equation:

En2 2−En1 2En−2δ0,n, forn∈Z. 1.10 Forf∈UDZp, the bosonicp-adic integral onZpis defined by

I1

f

Zp

fxdμ1x lim

N→ ∞

1 p^N

p^N−1 x0

fx, see4. 1.11

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From1.11, we can easily derive the followingI1-integral equation:

I1

f1

I f

f0, see4, 7, 8, 1.12

wheref1x fx1andf0 dfx/dx|_x0.

It is well known that the Bernoulli polynomial can be represented by the bosonicp-adic integral onZpas follows:

Zp

e^xytdμ1

y t

e^t−1e^xt^∞

n0Bnxtⁿ

n!, 1.13

whereBnxis called thenth Bernoulli polynomialsee4,7–13. In the special case,x0, Bn0 Bn is called thenth Bernoulli number. By the definition of Bernoulli numbers and polynomials, we get

Bnx

Zp

xy_n dμ1

y ⁿ

l0

n l

x^n−lBl. 1.14

Thus, by1.13and1.14, we see that

B01, B1ⁿ−Bnδ1,n, 1.15

with the usual convention about replacingBⁿbyBnsee1–22.

By1.11, we easily get

Zp

1−xy_n dμ1

y

−1ⁿ

Zp

xy_n dμ1

y

. 1.16

From1.13,1.14, and1.16, we have

Bn1−x −1ⁿBnx forn∈Z. 1.17

By1.15, we get

Bn2 nBn1 nBnδ1,n. 1.18

−1ⁿBn−1 Bn2 nBnδ1,n, see4. 1.19 From1.12and1.13, we get

Zp

x1y_n1 dμ1

y

−

Zp

xy_n1 dμ1

y

n1xⁿ. 1.20

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Bn1x1−Bn1x n1xⁿ forn∈Z. 1.21

Equation1.21is equivalent to the following equation:

xⁿ 1 n1

n l0

n1 l

Blx forn∈Z. 1.22

In this paper we derive some interesting and new identities for the Bernoulli and Euler numbers from thep-adic integral equations onZp.

2. Some Identities on Bernoulli and Euler Numbers

From1.1, we note that

Zp

1−xy_n dμ−1

y

−1ⁿ

Zp

xy_n dμ−1

y

. 2.1

By1.14and2.1, we get

En1−x −1ⁿEnx, wheren∈Z. 2.2

In the special case,x−1, we have

En2 −1ⁿEn−1 2En−2δ0,n. 2.3

Let us consider the following fermionicp-adic integral onZpas follows:

Zp

xⁿdμ−1x 1 n1

n l0

n1

l _Z_pBlxdμ−1x 1

n1 n

l0

n1 l

_l

k0

l k

Bl−k

Zp

x^kdμ−1x

1 n1

n l0

n1 l

_l

k0

l k

Bl−kEk.

2.4

Therefore, by1.4and2.4, we obtain the following theorem.

Theorem 2.1. Forn∈Z, one has En 1

n1 n

l0

n1 l

_l

k0

l k

B_l−kEk. 2.5

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It is known thatBnx −1ⁿBn1−x. If we take the fermionic p-adic integral on both sides of1.22, then we have

Zp

xⁿdμ−1x 1 n1

n l0

n1

l _Z_pBlxdμ₋₁x 1

n1 n

l0

n1 l

−1^l

Zp

Bl1−xdμ−1x

1 n1

n l0

n1 l

−1^l^l

k0

l k

B_l−k

Zp

1−x^kdμ₋₁x

1 n1

n l0

n1 l

−1^l^l

k0

l k

Bl−k−1^kEk−1.

2.6

From2.2and2.6, we note that

Zp

xⁿdμ−1x 1 n1

n l0

n1 l

−1^l^l

k0

l k

Bl−kEk2

1 n1

n l0

n1 l

−1^l^l

k0

l k

Bl−k2Ek−2δ0,k

1 n1

n l0

n1 l

−1^l

2Bl1 ^l

k0

l k

Bl−kEk−2Bl

1 n1

n l0

n1 l

−1^l _l

k0

l k

B_l−kEk2δ1,l .

2.7

Theorem 2.2. Forn∈Z, one has

En 1 n1

n l0

n1 l

−1^l _l

k0

l k

Bl−kEk2δ1,l . 2.8

Corollary 2.3. Forn∈N, one has

2En 1 n1

n l0

n1 l

−1^l _l

k0

l k

B_l−kEk . 2.9

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Let us take the bosonicp-adic integral on both sides of1.9as follows:

Zp

xⁿdμ1x

Zp

Enx 1 2

n−1 l0

n l

Elx dμ1x

ⁿ

l0

n l

En−l

Zp

x^ldμ1x 1 2

n−1

l0

n l

_l

k0

l k

El−k

Zp

x^kdμ1x

ⁿ

l0

n l

En−lBl1 2

n−1 l0

n l

l k0

l k

El−kBk.

2.10

Thus, by1.14and2.10, we obtain the following theorem.

Theorem 2.4. Forn∈Z, one has

Bnⁿ

l0

n l

E_n−lBl1 2

n−1

l0

n l

_l

k0

l k

E_l−kBk. 2.11

On the other hand, by2.2and2.10, we get

Zp

xⁿdμ1x −1ⁿ

Zp

En1−xdμ1x 1 2

n−1 l0

n l

−1^l

Zp

El1−xdμ1x

−1ⁿⁿ

l0

n l

E_n−l

Zp

1−x^ldμ1x

1 2

n−1

l0

n l

−1^l^l

k0

l k

El−k

Zp

1−x^kdμ1x

−1ⁿⁿ

l0

n l

En−l−1^lBl−1 1 2

n−1

l0

n l

−1^l^l

k0

l k

El−k−1^kBk−1

−1ⁿⁿ

l0

n l

E_n−lBl2 1 2

n−1 l0

n l

−1^l^l

k0

l k

E_l−kBk2

−1ⁿⁿ

l0

n l

E_n−llBlδ1,l

1 2

n−1

l0

n l

−1^l^l

k0

l k

E_l−kkBkδ1,k

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−1ⁿnE_n−11 −1ⁿⁿ

l0

n l

E_n−lBl −1ⁿnE_n−1 1 2

n−1

l0

n l

−1^llE_l−11

1 2

n−1

l0

n l

−1^l^l

k0

l k

El−kBk1 2

n−1

l1

n l

−1^llEl−1

−1ⁿn2En−1−2δ0,n−1 −1ⁿⁿ

l0

n l

En−lBl −1ⁿnEn−1

1 2

n−1

l0

n l

−1^ll2E_l−1−δ_0,l−1 1 2

n−1 l0

n l

−1^l^l

k0

l k

E_l−kBk

1 2

n−1

l1

n l

−1^llE_l−1,

2.12

wheren∈Nwithn≥2. Therefore, by2.12, we obtain the following theorem.

Theorem 2.5. Forn∈Nwithn≥2, one has

B_2n−1−2n−1

2 −2n−1E_2n−2−1−²ⁿ⁻¹

l0

2n−1 l

E_2n−1−lBl

1 2

2n−2

l0

2n−1 l

−1^l^l

k0

l k

El−kBk.

2.13

By1.9and1.22, we get

Zp

x^myⁿdμ−1xdμ1

y

Zp

1 m1

m k0

m1 k

Bkx En

y 1

2

n−1

l0

n l

El

y

dμ−1xdμ1

y

1 m1

m k0

m1 k

Zp

BkxEn

y

dμ₋₁xdμ1

y

1

2m1

m k0

n−1 l0

m1 k

n l

Zp

BkxEl

y

dμ−1xdμ1

y

1 m1

m k0

k l0

n p0

m1 k

k l

n p

B_k−lE_n−pBpEl

1

2m1

m k0

n−1 l0

k s0

l p0

m1 k

n l

k s

l p

B_k−sE_l−pEsBp.

2.14

Therefore, by1.4,1.14, and2.14, we obtain the following theorem.

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Theorem 2.6. Form∈Zandn∈N, one has EmBn 1

m1 m k0

k l0

n p0

m1 k

k l

n p

Bk−lEn−pBpEl

1

2m1

m k0

n−1

l0

k s0

l p0

m1 k

n l

k s

l p

B_k−sE_l−pEsBp.

2.15

It is easy to show that

Zp

x^mndμ−1x

Zp

1 m1

m k0

m1 k

Bkx Enx 1 2

n−1

l0

n l

Elx dμ−1x 1

m1 m k0

k i0

n j0

m1 k

k i

n j

Bk−iEn−j

Zp

x^ijdμ−1x 1

2m1

m k0

n−1

l0

k i0

l j0

m1 k

n l

k i

l j

Bk−iEl−j

Zp

x^ijdμ−1x 1

m1 m k0

k i0

n j0

m1 k

k i

n j

Bk−iEn−jEij

1

2m1

m k0

n−1

l0

k i0

l j0

m1 k

n l

k i

l j

Bk−iEl−jEij.

2.16

Therefore, by2.16, we obtain the following corollay.

Corollary 2.7. Form∈Zandn∈N, one has

E_mn 1 m1

m k0

k i0

n j0

m1 k

k i

n j

B_k−iE_n−jE_ij

1

2m1

m k0

n−1

l0

k i0

l j0

m1 k

n l

k i

l j

Bk−iEl−jEij.

2.17

Forf∈CZp, p-adic analogue of Bernstein operator of ordernforfis given by

Bn

f|x ⁿ

k0

f k

n n

k

x^k1−x^n−kⁿ

k0

f k

n

Bk,nx, 2.18

whereBk,nx ⁿkx^k1−x^n−kforn, k ∈Zis called the Bernstein polynomial of degreen see8. From the definition ofBk,nx, we note thatB_n−k,n1−x Bk,nx.

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Let us take the fermionicp-adic integral onZp for the product ofx^m and Bk,nxas follows:

Zp

x^mBk,nxdμ−1x 1 m1

m l0

m1

l _Z_pBlxBk,nxdμ−1x ⁿ_k

m1 m

l0

l j0

m1 l

l j

B_l−j

Zp

x^jk1−x^n−kdμ₋₁x

ⁿ_k m1

m l0

l j0

n−k

i0

−1ⁱBl−j

m1 l

l j

n−k

i _Z_px^ijkdμ−1x

ⁿ_k m1

m l0

l j0

n−k

i0

−1ⁱ m1

l l

j

n−k i

Bl−jEijk.

2.19

From2.18, we note that

Zp

x^mBk,nxdμ−1x n

k _Z_px^mk1−x^n−kdμ−1x

n k

_n−k

j0

n−k j

−1^j

Zp

x^mkjdμ₋₁x

n

k _n−k

j0

n−k j

−1^jEmkj.

2.20

Theorem 2.8. Form, n, k∈Z, one has

n−k

j0

n−k j

−1^jEmkj 1 m1

m l0

l j0

n−k

i0

−1ⁱ m1

l l

j

n−k i

Bl−jEijk. 2.21

In particular,

m1Emn^m

l0

l j0

m1 l

l j

Bl−jEjn. 2.22

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By1.17and the symmetric property ofBk,nx, we get

Zp

x^mBk,nxdμ₋₁x

Zp

x^mB_n−k,n1−xdμ₋₁x 1

m1 m

l0

−1^l m1

l _Z_pBl1−xB_n−k,n1−xdμ₋₁x ⁿ_k

m1 m

l0

l j0

k i0

−1^il m1

l l

j k

i

Bl−j

Zp

1−x^ijn−kdμ−1x.

2.23

From1.4and2.2, we note that

Zp

1−xⁿdμ₋₁x −1ⁿEn−1 En2 2En−2δ0,n. 2.24

By2.23and2.24, we see that

Zp

x^mBk,nxdμ₋₁x ⁿ_k m1

m l0

l j0

k i0

−1^il m1

l l

j k

i

B_l−j

2E_ijn−k−2δ_0,ijn−k . 2.25

From2.20and2.25, we have

n−k

j0

n−k j

−1^jEmkj

2δ0,k

m1 m

l0

l j0

−1^l m1

l l

j

Bl−j− 2 m1

m l0

−1^l m1

l

Blδk,n

1 m1

m l0

l j0

k i0

−1^il m1

l l

j k

i

Bl−jEijn−k

2δ0,k

m1 m

l0

l j0

−1^l m1

l l

j

Bl−j− 2 m1

Bm12 −1^mBm1 δk,n

1 m1

m l0

l j0

k i0

−1^il m1

l l

j k

i

B_l−jE_ijn−k.

2.26

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Theorem 2.9. Form, n, k∈Nwithn≥k, one has

n−k

j0

n−k j

−1^jEmkj 1 m1

m l0

l j0

k i0

−1^il m1

l l

j k

i

Bl−jEijn−k

− 2 m1

B_m1m1 −1^mB_m1 .

2.27

In particular,

2m2E2mn12 ^2m1

l0

l j0

n i0

−1^il

2m2 l

l j

n i

Bl−jEij. 2.28

Acknowledgment

The first author was supported by National Research Foundation of Korea Grant funded by the Korean Government 2011-0002486.

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