Integral Formulae of Bernoulli and Genocchi Polynomials

(1)

Volume 2012, Article ID 472010,8pages doi:10.1155/2012/472010

Research Article

Integral Formulae of Bernoulli and Genocchi Polynomials

Seog-Hoon Rim,

¹

Joung-Hee Jin,

²

and Joohee Jeong

¹

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

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

Correspondence should be addressed to Seog-Hoon Rim,[email protected] Received 19 June 2012; Accepted 19 July 2012

Academic Editor: Taekyun Kim

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.

Recently, some interesting and new identities are introduced in the work of Kim et al.2012. From these identities, we derive some new and interesting integral formulae for Bernoulli and Genocchi polynomials.

1. Introduction

As it is well known, the Bernoulli polynomials are defined by generating functions as follows:

t

e^t−1e^xte^Bxt^∞

n0Bnxtⁿ

n! 1.1

see1–5 with the usual convention about replacing BⁿxbyBnx. In the special case, x0,Bn0 Bnare called thenth Bernoulli numbers.

The Genocchi polynomials are also defined by

2t

e^t1e^xte^Gxt^∞

n0

Gnxtⁿ

n! 1.2

see1,6–10with the usual convention about replacingGⁿxbyGnx. In the special case, x0,Gn0 Gnare called thenth Genocchi numbers.

(2)

From1.1, we note that

Bnx ⁿ

l0

n l

Blx^n−l 1.3

see1–5. Thus, by1.3, we get

d

dxBnx n n−1

l0

n−1 l

Blx^n−1−lnBn−1x 1.4

see2. From1.2, we note that

Gnx ⁿ

l0

n l

Glx^n−l. 1.5

From1.5, we can derive the following equation:

d

dxGnx n

n−1

l0

n−1 l

Glx^n−1−lnGn−1x. 1.6

By the definition of Bernoulli and Genocchi numbers, we get the following recurrence formulae:

B01, Bn1−Bnδ1,n, G00, Gn1 Gn2δ1,n, 1.7

whereδn,kis the Kronecker symbolsee2. From1.4,1.6, and1.7, we note that

₁

0

Bnxdx δ0,n

n1 n≥0,

₁

0

Gnxdx−2G_n1

n1 n≥1. 1.8

From the identities of Bernoulli and Genocchi polynomials, we derive some new and interesting integral formulae of an arithmetical nature on the Bernoulli and Genocchi polynomials.

(3)

2. Integral Formula of Bernoulli and Genocchi Polynomials

From1.1and1.2, we note that

t

e^t−1e^xt 1 2

2te^xt e^t1

1

t t

e^t−1

2te^xt e^t1

1 2

_∞

n0

Gnxtⁿ n!

1

t _∞

l0

Blt^l l!

_∞

m0

Gmxt^m m!

1 2

∞ n0

Gnxtⁿ n!1

t ∞ n0

n l0

n l

GlxB_n−ltⁿ n!

1 2

∞ n0

Gnxtⁿ n!^∞

n0

⎛

⎜⎜

⎝−1

2Gnx ⁿ¹

l /l0n

_n1

l

GlxB_n1−l n1

⎞

⎟⎟

⎠tⁿ n!

^∞

n0

⎛

⎜⎜

⎝

n1

l /l0n

n1 l

GlxBn1−l

n1

⎞

⎟⎟

⎠tⁿ n!.

2.1

By comparing the coeﬃcients on the both sides of2.1, we obtain the following theorem.

Theorem 2.1. Forn∈Z, one has

Bnx ⁿ¹

l0l /n

n1 l

GlxBn1−l

n1 . 2.2

From1.1and1.2, also notes that

2t

e^t1e^xt 1 t

2t e^t−1 e^t1

te^xt e^t−1

1

t

2t−2 2t e^t1

te^xt e^t−1

1 t

2t−2

∞ l0

Glt^l l!

_∞

m0

Bmxt^m m!

1 t

−2^∞

l1

Gl1

l1 t^l1

l!

_∞

m0

Bmxt^m m!

^∞

n1

−2ⁿ

l1

n l

Gl1

l1Bn−lx tⁿ

n!.

2.3

By comparing the coeﬃcients on the both sides of2.3, we obtain the following theorem.

(4)

Theorem 2.2. Forn∈N, one has

Gnx −2ⁿ

l1

n l

Gl1

l1Bn−lx. 2.4

Let one take the definite integral from 0 to 1 on both sides ofTheorem 2.1. Forn≥2,

0−2ⁿ¹

l1l /n

n1 l

G_l1 l1

B_n1−l

n1 −BnG2−2 n l /l1n−1

n l

B_n−lG_l2

l1l2. 2.5

Therefore, by2.3, we obtain the following theorem.

Theorem 2.3. Forn∈Nwithn≥2, one has

Bn2 n l /l1n−1

n l

Bn−lGl2

l1l2. 2.6

3. p -Adic Integral on Z

p

Associated with Bernoulli and Genocchi Numbers

Letpbe a fixed odd prime number. Throughout this section,Zp,Qp, andCpwill denote the ring ofp-adic integers, the field ofp-adic rational numbers, and the completion of algebraic closure ofQp, respectively. Letvpbe the normalized exponential valuation ofCpwith|p|p p^−v^p^p 1/p. LetUDZpbe the space of uniformly diﬀerentiable functions onZp. Forf ∈ UDZp, the bosonicp-adic integral onZpis defined by

I f

Zp

fxdμx lim

N→ ∞

1 p^N

p^N−1 x0

fx 3.1

see2,5,11. From3.1, we can derive the following integral equation:

I fn

I f

ⁿ⁻¹

i0

fi n∈N, 3.2

wherefnx fxnandfi dfx/dx|_xisee2. Let us takefy e^txy. Then we have

Zp

e^txydμ y

t

e^t−1e^xt^∞

n0

Bnxtⁿ

n! 3.3

(5)

see2,5. From3.3, we have

Zp

xnⁿdμ y

Bnx,

Zp

yⁿdμ y

Bn 3.4

see2,5. Thus, by3.2and3.4, we get

Zp

xn^mdμx

Zp

x^mdμx m

n−1

i0

i^m−1, 3.5

see2. From3.5, we have

Bmn−Bmm n−1

i0

i^m−1 n∈Z 3.6

see2. The fermionicp-adic integral onZpis defined by Kim as follows2,8,9:

I₋₁ f

Zp

fxdμ₋₁x lim

N→ ∞

1 p^N

p^N−1 x0

fx−1^x. 3.7

From3.7, we obtain the following integral equation:

I₋₁ fn

−1ⁿI₋₁ f

2

n−1

l0

−1^n−l−1fl 3.8

see2, wherefnx fxn. Thus, by3.8, we have

Zp

xn^mdμ−1x −1ⁿ

Zp

x^mdμ−1x 2

n−1

l0

−1^n−l−1l^m 3.9

see2. Let us takefy e^txy. Then we have

t

Zp

e^txydμ₋₁ y

2te^xt e^t1 ^∞

n0

Gnxtⁿ

n!. 3.10

From3.10, we have

Zp

xy_n dμ−1

y

Gn1x n1 ,

Zp

yⁿdμ−1 y

Gn1

n1. 3.11

(6)

Thus, by3.9and3.11, we get

Gm1n

m1 −1ⁿ Gn1

n12 n−1

l0

−1^l−1l^m

. 3.12

Let us consider the followingp-adic integral onZp:

K1

Zp

Bnxdμx ⁿ

l0

n l

B_n−l

Zp

x^ldμx ⁿ

l0

n l

B_n−lBl. 3.13

FromTheorem 2.1and3.13, one has

K1 ⁿ¹

k /k0n

n1 k

B_n1−k n1

k l0

k l

G_k−l

Zp

x^ldμx

ⁿ¹

k /k0n

k l0

n1 k

k l

B_n1−kBlG_k−l n1 .

3.14

Therefore, by3.13and3.14, we obtain the following theorem.

n l0

n l

Bn−lBl ⁿ¹

k /k0n

k l0

n1 k

k l

Bn1−kBlGk−l

n1 . 3.15

Now, one sets

K2

Zp

Bnxdμ−1x ⁿ

l0

n l

Bn−lGl1

l1. 3.16

ByTheorem 2.1, one gets

K2 ⁿ¹

k /k0n

n1 k

Bn1−k

n1 k

l0

k l

Gk−l

Zp

x^ldμ−1x

ⁿ¹

k0k /n

k l0

n1 k

k l

Bn1−kGk−lGl1

n1l1 .

3.17

(7)

n l0

n l

Bn−lGl1

l1 ⁿ¹

k /k0n

k l0

n1 k

k l

Bn1−kGk−lGl1

n1l1 . 3.18

Let us consider the followingp-adic integral onZp:

K3

Zp

Gnxdμ−1x ⁿ

l0

n l

Gn−l

Zp

x^ldμ−1x ⁿ

l0

n l

Gn−lGl1

l1. 3.19

FromTheorem 2.2, one has

K3−2ⁿ

l1

n l

Gl1

l1 n−l k0

n−l k

Bn−l−k

Zp

x^kdμ−1x

−2ⁿ

l1

n−l k0

n l

n−l k

B_n−l−k Gl1Gk1

l1k1.

3.20

n l0

n l

Gn−lGl1

l1 −2ⁿ

l1

n−l k0

n l

n−l k

Bn−l−kGl1Gk1

l1k1 . 3.21

Now, one sets

K4

Zp

Gnxdμx ⁿ

l0

n l

Gn−lBl. 3.22

ByTheorem 2.2, one gets

K4−2ⁿ

l1

n−l k0

n l

n−l k

Gl1

l1Bn−l−kBk. 3.23

Therefore, by3.22and3.23, we obtain the following corollary.

Corollary 3.4. Forn∈Z, one has

n l0

n l

G_n−lBl−2ⁿ

l1

n−l k0

n l

n−l k

G_l1B_n−l−kBk

l1 . 3.24

(8)

Acknowledgement

This research was supported by Kyungpook National University research Fund, 2012.

References

1 A. Bayad and T. Kim, “Identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials,” Advanced Studies in Contemporary Mathematics, vol. 20, no. 2, pp. 247–253, 2010.

2 D. S. Kim, D. V. Dolgy, H. M. Kim, S. H. Lee, and T. Kim, “Integral formulae of Bernoulli polynomials,”

Discrete Dynamics in Nature and Soceity, vol. 2012, Article ID 269847, 15 pages, 2012.

3 T. Kim, “On the weightedq-Bernoulli numbers and polynomials,” Advanced Studies in Contemporary Mathematics, vol. 21, no. 2, pp. 207–215, 2011.

4 H. Ozden, I. N. Cangul, and Y. Simsek, “Remarks onq-Bernoulli numbers associated with Daehee numbers,” Advanced Studies in Contemporary Mathematics, vol. 18, no. 1, pp. 41–48, 2009.

5 S.-H. Rim, E.-J. Moon, S.-J. Lee, and J.-H. Jin, “Multivariate twistedp-adicq-integral onZpassociated with twistedq-Bernoulli polynomials and numbers,” Journal of Inequalities and Applications, vol. 2010, Article ID 579509, 6 pages, 2010.

6 I. N. Cangul, V. Kurt, H. Ozden, and Y. Simsek, “On the higher-orderw-q-Genocchi numbers,”

Advanced Studies in Contemporary Mathematics, vol. 19, no. 1, pp. 39–57, 2009.

7 I. N. Cangul, H. Ozden, and Y. Simsek, “A new approach to q-Genocchi numbers and their interpolation functions,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 71, no. 12, pp. e793–

e799, 2009.

8 T. Kim, “A note on theq-Genocchi numbers and polynomials,” Journal of Inequalities and Applications, Article ID 71452, 8 pages, 2007.

9 T. Kim, “On the multipleq-Genocchi and Euler numbers,” Russian Journal of Mathematical Physics, vol.

15, no. 4, pp. 481–486, 2008.

10 S.-H. Rim, E.-J. Moon, S.-J. Lee, and J.-H. Jin, “On the symmetric properties for the generalized Genocchi polynomials,” Journal of Computational Analysis and Applications, vol. 13, no. 7, pp. 1240–

1245, 2011.

11 T. Kim, “q-Volkenborn integration,” Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288–299, 2002.

(9)

Submit your manuscripts at http://www.hindawi.com

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Mathematics

^{Journal of}

Hindawi Publishing Corporation http://www.hindawi.com

Differential Equations

International Journal of

Volume 2014

Applied Mathematics^{Journal of}

Mathematical PhysicsAdvances in

Complex Analysis

^{Journal of}

Optimization

^{Journal of}

Combinatorics

Journal of

Function Spaces

Abstract and Applied Analysis

International Journal of Mathematics and Mathematical Sciences

The Scientific World Journal

Discrete Dynamics in Nature and Society

Discrete Mathematics

^{Journal of}

Integral Formulae of Bernoulli and Genocchi Polynomials

Research Article