Identities on the Bernoulli and

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Volume 2012, Article ID 184649,9pages doi:10.1155/2012/184649

Research Article

Identities on the Bernoulli and

Genocchi Numbers and Polynomials

Seog-Hoon Rim,

¹

Joohee Jeong,

¹

and Sun-Jung Lee

²

1Department of Mathematics Education, Kyungpook National University, Taegu 702-701, Republic of Korea

2Department of Mathematics, Kyungpook National University, Daegu 702-701, Republic of Korea

Correspondence should be addressed to Seog-Hoon Rim,[email protected] Received 9 June 2012; Accepted 9 August 2012

Academic Editor: Yilmaz Simsek

Copyrightq2012 Seog-Hoon Rim 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.

We give some interesting identities on the Bernoulli numbers and polynomials, on the Genocchi numbers and polynomials by using symmetric properties of the Bernoulli and Genocchi polynomials.

1. Introduction

Letp be a fixed odd prime number. Throughout this paperZp,Qp, and Cp will denote the ring ofp-adic rational integers, the field ofp-adic rational numbers, and the completion of the algebraic closure ofQp. LetNbe the set of natural numbers andZN∪ {0}. Thep-adic norm onCpis normalized so that|p|p p⁻¹. LetCZpbe the space of continuous functions onZp. Forf∈CZp, the fermionicp-adic integral onZpis defined by Kim as follows:

I₋₁ f

Zp

fxdμ₋₁x lim

N→ ∞ p^N−1

x0

fx−1^x 1.1

see1–16. From1.1, we have I−1

f1

−I−1 f

2f0 1.2

see1–16, wheref1x fx1.

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Let us takefx e^xt. Then, by1.2, we get

t

Zp

e^xtdμ₋₁x 2t e^t1 ^∞

n0

Gntⁿ

n!, 1.3

whereGnare thenth ordinary Genocchi numberssee8,15.

From the same method of1.3, we can also derive the following equation:

t

Zp

e^xytdμ₋₁ y

2t

e^t1e^xt^∞

n0

Gnxtⁿ

n!, 1.4

whereGnxare called thenth Genocchi polynomialssee14,15.

By1.3, we easily see that

Gnx ⁿ

l0

n l

Glx^n−l 1.5

see15. By 1.3 and 1.4, we get Witt’s formula for the nth Genocchi numbers and polynomials as follows:

Zp

xⁿdμ₋₁x Gn1

n1,

Zp

xyn

dμ₋₁ y

Gn1x

n1 , forn∈Z. 1.6

From1.2, we have

Zp

x1ⁿdμ₋₁x

Zp

xⁿdμ₋₁x 2δ0,n, 1.7

where the symbolδ0,nis the Kronecker symbolsee4,5.

Thus, by1.5and1.7, we get

G1ⁿGn2δ1,n 1.8

see15. From1.4, we can derive the following equation:

Zp

1−xy_n dμ−1

y

−1ⁿ

Zp

xy_n dμ−1

y

. 1.9

By1.6and1.9, we see that

Gn11−x

n1 −1ⁿGn1x

n1 . 1.10

Thus, by1.10, we getG_n12/n1 −1ⁿG_n1−1/n1.

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From1.5and1.8, we have Gn12

n1 2−Gn11

n1 2 Gn1

n1 −2δ1,n1. 1.11

The Bernoulli polynomialsBnxare defined by

t

e^t−1e^xte^Bxt^∞

n0

Bnxtⁿ

n! 1.12

see6,9,12with the usual convention about replacingBⁿxbyBnx.

In the special case,x0,Bn0 Bnis called then-th Bernoulli number. By1.12, we easily see that

Bnx ⁿ

l0

n l

x^n−lBl Bxⁿ 1.13

see6. Thus, by1.12and1.13, we get reflection symmetric formula for the Bernoulli polynomials as follows:

Bn1−x −1ⁿBnx, 1.14

B01, B1ⁿ−Bnδ1,n 1.15

see6,9,12. From1.14and1.15, we can also derive the following identity:

−1ⁿBn−1 Bn2 nBn1 nBnδ1,n. 1.16

In this paper, we investigate some properties of the fermionicp-adic integrals onZp. By using these properties, we give some new identities on the Bernoulli and the Euler numbers which are useful in studying combinatorics.

2. Identities on the Bernoulli and Genocchi Numbers and Polynomials

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

I1

Zp

Bnxdμ−1x ⁿ

l0

n l

Bn−l

Zp

x^ldμ−1x

ⁿ

l0

n l

B_n−lG_l1

l1, forn∈ZN∪ {0}.

2.1

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On the other hand, by1.14and1.15, we get

I1 −1ⁿ

Zp

Bn1−xdμ−1x

−1ⁿⁿ

l0

n l

Bn−l

Zp

1−x^ldμ−1x

−1ⁿⁿ

l0

n l

Bn−l−1^lG_l1−1 l1 −1ⁿⁿ

l0

n l

Bn−l

2 Gl1

l1 −2δ1,l1

2−1ⁿBnδ1,n −1ⁿⁿ

l0

n l

Bn−lGl1

l12−1ⁿ¹Bn.

2.2

Equating2.1and2.2, we obtain the following theorem.

Theorem 2.1. Forn∈Z, one has

1 −1ⁿ¹ ⁿ

l0

n l

B_n−lG_l1

l1 2−1ⁿδ1,n. 2.3

By using the reflection symmetric property for the Euler polynomials, we can also obtain some interesting identities on the Euler numbers.

Now, we consider the fermionicp-adic integral onZpfor the polynomials as follows:

I2

Zp

Gnxdμ₋₁x

ⁿ

l0

n l

Gn−l

Zp

x^ldμ−1x

ⁿ

l0

n l

Gn−lGl1

l1, forn∈Z.

2.4

On the other hand, by1.8,1.10, and1.11, we get

I2 −1ⁿ⁻¹

Zp

Gn1−xdμ−1x

−1ⁿ⁻¹ⁿ

l0

n l

Gn−l

Zp

1−x^ldμ−1x

−1ⁿ⁻¹ⁿ

l0

n l

G_n−l−1^lG_l1−1 l1

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−1ⁿ⁻¹ⁿ

l0

n l

Gn−l

2Gl1

l1−2δ1,l1

2−1ⁿ⁻¹2δ1,n−Gn 2−1ⁿGn

−1ⁿ⁻¹ⁿ

l0

n l

Gn−lGl1

l1.

2.5

Equating2.4and2.5, we obtain the following theorem.

Theorem 2.2. Forn∈Z, one has

1 −1ⁿⁿ

l0

n l

G_n−lG_l1

l1 4−1ⁿGn4−1ⁿ¹δ1,n. 2.6

Let us consider the fermionicp-adic integral onZpfor the product ofBnxandGnx as follows:

I3

Zp

BmxGnxdμ−1x

^m

k0

n l0

m k

n l

Bm−kGn−l

Zp

x^kldμ−1x

^m

k0

n l0

m k

n l

B_m−kG_n−l G_kl1 kl1.

2.7

I3

Zp

BmxGnxdμ−1x −1^nm−1

Zp

Bm1−xGn1−xdμ−1x

−1^nm−1^m

k0

n l0

m k

n l

Bm−kGn−l

Zp

1−x^kldμ−1x

2−1^nm−1Bm1Gn1 2−1^mnBmGn

−1^nm−1^m

k0

n l0

m k

n l

B_m−kG_n−l G_kl1 kl1.

2.8

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By2.7and2.8, we easily see that

1 −1^nm1 ^m

k0

n l0

m k

n l

Bm−kGn−l Gkl1

kl1

2−1^mn−1δ1,mBm2δ1,n−Gn 2−1^mnBmGn

4−1^mn−1Bmδ1,n2−1^mnBmGn4−1^mn−1δ1,mδ1,n

2−1^mnδ1,mGn2−1^mnBmGn.

2.9

Therefore, by2.9, we obtain the following theorem.

Theorem 2.3. Forn, m∈Z, one has

1 −1^nm1 ^m

k0

n l0

m k

n l

B_m−k G_n−l1 n−l1

G_kl1 kl1

4−1^mnBmGn4−1^mn−1Bmδ1,n4−1^mn−1δ1,mδ1,n

2−1^mnδ1,mGn.

2.10

Corollary 2.4. Forn, m∈N, one has

2m k0

2n l0

2m k

2n l

B2m−kG2n−l Gkl1

kl1 2B2mG2n. 2.11

Let us consider the fermionicp-adic integral on Zp for the product of the Bernoulli polynomials and the Bernstein polynomials. For n, k ∈ Z, with 0 ≤ k ≤ n, Bk,nx ⁿ_kx^k1−x^n−kare called the Bernstein polynomials of degreen, see11. It is easy to show thatBk,nx Bn−k,n1−x,

I4

Zp

BmxBk,nxdμ₋₁x

n k

_m

l0

m l

B_m−l

Zp

x^kl1−x^n−kdμ₋₁x

n

k _m

l0 n−k

j0

m l

n−k j

−1^jBm−l

Zp

x^kljdμ−1x

n

k _m

l0 n−k

j0

m l

n−k j

−1^jBm−l G_klj1 klj1.

2.12

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I4 −1^m

Zp

Bm1−xB_n−k,n1−xdμ₋₁x

−1^m n

k _m

l0

k j0

m l

k j

−1^jBm−l

Zp

1−x^n−kljdμ−1x −1^m

n k

m l0

k j0

m l

k j

−1^jBm−l

×

2−2δ1,n−klj1 Gn−klj1

n−klj1

2−1^m n

k

Bm1δ0,k2−1^m1 n

k

Bmδk,n

−1^m n

k _m

l0

k j0

m l

k j

−1^jB_m−l G_n−klj1 n−klj1.

2.13

Equating2.12and2.13, we see that m

l0 n−k

j0

m l

n−k j

−1^jBm−l G_klj1 klj1 2−1^mBm1δ0,k2−1^m1Bmδk,n

−1^m^m

l0

k j0

m l

k j

−1^jBm−l Gn−klj1

n−klj1.

2.14

Thus, from2.14, we obtain the following theorem.

Theorem 2.5. Forn, m∈N, one has 2m

l0

n j0

2m l

n j

−1^jB2m−l Glj1

lj1 2B2m1 ^2m

l0

2m l

B2m−l Gnl1

nl1. 2.15

Finally, we consider the fermionicp-adic integral on Zpfor the product of the Euler polynomials and the Bernstein polynomials as follows:

I5

Zp

GmxBk,nxdμ₋₁x

n k

_m

l0

m l

G_m−l

Zp

x^kl1−x^n−kdμ₋₁x

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n k

m l0

n−k

j0

m l

n−k j

−1^jGm−l

Zp

x^kljdμ−1x

n

k _m

l0 n−k

j0

m l

n−k j

−1^jG_m−l G_klj1 klj1.

2.16

I5 −1^m−1

Zp

Gm1−xBn−k,n1−xdμ−1x

−1^m−1 n

k _m

l0

m l

Gm−l

k j0

k j

−1^j

Zp

1−x^n−kljdμ−1x −1^m−1

n k

_m

l0

k j0

m l

k j

−1^jGm−l

×

2 G_n−klj1

n−klj1−2δ_1,n−klj1

2−1^m−1 n

k

Gm1δ0,k2−1^m n

k

Gmδk,n

−1^m−1 n

k m

l0

k j0

m l

k j

−1^jGm−l Gn−klj1

n−klj1.

2.17

Equating2.16and2.17, we obtain

m l0

n−k

j0

m l

n−k j

−1^jGm−l Gklj1

klj1 2−1^m−1Gm1δ0,k2−1^mGmδk,n

−1^m−1^m

l0

k j0

m l

k j

−1^jGm−l Gn−klj1

n−klj1.

2.18

Therefore, by2.18, we obtain the following theorem.

Theorem 2.6. Forn, m∈N, one has 2m

l0

n j0

2m l

n j

−1^jG2m−l G_lj1

lj1 −2G2m1−^2m

l0

2m l

G2m−l Gnl1

nl1. 2.19

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Acknowledgment

This paper was supported by Kynugpook National University Research Fund, 2012.

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Identities on the Bernoulli and

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