Some New Identities on the Bernoulli and Euler Numbers

(1)

Volume 2011, Article ID 856132,11pages doi:10.1155/2011/856132

Research Article

Some New Identities on the Bernoulli and Euler Numbers

Dae San Kim,

¹

Taekyun Kim,

²

Sang-Hun Lee,

³

D. V. Dolgy,

⁴

and Seog-Hoon Rim

⁵

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, Kwangwoon University, Seoul 139-701, Republic of Korea

4Hanrimwon, Kwangwoon University, Seoul 139-701, Republic of Korea

5Department of Mathematics Education, Kyungpook National University, Taegu 702-701, Republic of Korea

Correspondence should be addressed to Taekyun Kim,[email protected] Received 6 October 2011; Revised 31 October 2011; Accepted 31 October 2011 Academic Editor: Lee-Chae Jang

Copyrightq2011 Dae San 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.

We give some new identities on the Bernoulli and Euler numbers by using the bosonic p-adic integral onZpand reflection symmetric properties of Bernoulli and Euler polynomials.

1. Introduction

Letpbe a fixed prime number. Throughout this paperZp,Qp, andCpwill denote the ring of p-adic rational integers, the field ofp-adic rational numbers, and the completion of algebraic closure ofQp. Let UDZpbe the space of uniformly diﬀerentiable functions onZp. Forf ∈ UDZp, the bosonicp-adic integral onZpis defined by

I f

Zp

fxdμx lim

N→ ∞ p^N−1

x0

fxμ

xp^NZp

lim

N→ ∞

1 p^N

p^N−1 x0

fx. 1.1

From1.1, we note that

I f1

I f

f0, wheref1x fx1, 1.2

(2)

see1. As is well known, the ordinary Bernoulli polynomials are defined by the generating function as follows:

Ft, x t

e^t−1e^xte^Bxt^∞

n0

Bnxtⁿ

n!, 1.3

see 1–19, where we use the technical notation by replacing Bⁿx by Bnxn ≥ 0, symbolically. In the special case, x 0, Bn0 Bn are called the n-th ordinary Bernoulli numbers. That is, the generating function of ordinary Bernoulli numbers is given by

Ft Ft,0 t

e^t−1 ^∞

n0

Bntⁿ

n!, 1.4

see1–19. From1.4, we can derive the following relation:

B0 1, B1ⁿ−Bnδ1,n, 1.5 see1,10, whereδ1,nis the Kronecker symbol.

By1.3and1.4, we easily get

Bnx ⁿ

l0

n

l Blx^n−lⁿ

l0

n

l B_n−lx^l. 1.6

By1.2and1.3, we easily get

Zp

e^xytdμ y

t

e^t−1e^xt^∞

n0

Bnxtⁿ

n!, 1.7

see1,10. From1.7, we can derive Witt’s formula for then-th Bernoulli polynomials as follows:

Zp

xyn

dμ y

Bnx, where n∈Z, 1.8

see11. By1.1and1.8, we easily see that

Zp

y1−x_n dμ

y

−1ⁿ

Zp

yx_n dμ

y

. 1.9

Thus, by1.8and1.9, we get reflection symmetric relation for the Bernoulli polynomials as follows:

Bn1−x −1ⁿBnx wheren∈Z. 1.10

(3)

The ordinary Euler polynomials are defined by the generating function as follows:

Fet, x 2

e^t1e^xt^∞

n0

Enxtⁿ

n!. 1.11

with the usual convention about replacingEⁿxbyEnx see8,9. In the special case, x0,En0 Enare called then-th Euler numberssee8,9.

From1.11, we note that 2

e^t1e^xt 2

1e^−te^−1−xt^∞

n0

−1ⁿEn1−xtⁿ

n! , 1.12

By comparing the coeﬃcients on both sides of 1.11 and 1.12, we obtain the following reflection symmetric relation for Euler polynomials as follows:

Enx −1ⁿEn1−x, wheren∈Z. 1.13 The equations1.10and1.13are useful in deriving our main results in this paper.

Forn, k∈Z, the Bernstein polynomials are defined by

Bk,nx n

k x^k1−x^n−k, 1.14

see13. By1.14, we easily getBk,nx B_n−k,n1−x.

In this paper we consider thep-adic integrals for the Bernoulli and Euler polynomials.

From those p-adic integrals, we derive some new identities on the Bernoulli and Euler numbers.

2. Identities on the Bernoulli and Euler Numbers

First, we consider thep-adic integral onZp for thenth ordinary Bernoulli polynomials as follows:

I1

Zp

Bnxdμx ⁿ

l0

n l Bn−l

Zp

x^ldμx

ⁿ

l0

n

l Bn−lBl, wheren∈Z.

2.1

On the other hand, by1.3and1.10, one gets

I1 −1ⁿ

Zp

Bn1−xdμx. 2.2

(4)

From1.5,1.6,1.8, and2.2, one notes that

I1 −1ⁿⁿ

l0

n l B_n−l

Zp

1−x^ldμx

−1ⁿⁿ

l0

n

l B_n−llBlδ1,l

−1ⁿnBn−l1 −1ⁿⁿ

l0

n

l Bn−lBl −1ⁿnBn−l.

2.3

Equating2.1and2.3, one gets

1 −1ⁿ¹ⁿ

l0

n

l B_n−lBl −1ⁿnδ_1,n−lB_n−1 −1ⁿnB_n−1 2−1ⁿnB_n−l −1ⁿnδ_1,n−1.

2.4

Letn∈Nwithn≡1mod 2. Then, by2.4, one has

2n−1

l0

2n−1

l B2n−1−lBl−2n−1B2n−2. 2.5

Therefore, by2.4and2.5, we obtain the following theorem.

Theorem 2.1. Forn∈N, one has

1 −1ⁿ¹ⁿ

l0

n

l Bn−lBl2−1ⁿnBn−1 −1ⁿnδ1,n−1. 2.6

In particular,

2n−1

l0

2n−1

l B_2n−1−lBl−2n−1B_2n−2. 2.7

By the same motivation, let us also consider thep-adic integral onZpfor Euler polynomials as follows:

I2

Zp

Enxdμx ⁿ

l0

n l En−l

Zp

x^ldμx

ⁿ

l0

n

l En−lBl, wheren∈Z.

2.8

(5)

I2 −1ⁿ

Zp

En1−xdμx −1ⁿⁿ

l0

n l E_n−l

Zp

1−x^ldμx

−1ⁿⁿ

l0

n

l En−llBlδ1,l

n−1ⁿEn−l1 −1ⁿⁿ

l0

n

l En−lBl −1ⁿnEn−l.

2.9

From1.12and the definition of Euler numbers, one has

Enx ⁿ

l0

n

l Elx^n−lⁿ

l0

n

l En−lx^l Exⁿ, 2.10

E01, E1ⁿEn2δ0,n, 2.11 see8,9with the usual convention of replacingEⁿbyEn. By2.9,2.10, and2.11, one gets

I2 n−1ⁿ2δ0,n−1−En−1 −1ⁿnEn−1 −1ⁿⁿ

l0

n

l En−lBl. 2.12

Equating2.8and2.12, one has

1 −1ⁿ⁻¹ⁿ

l0

n

l En−lBl2n−1ⁿδ0,n−1. 2.13

Therefore, by2.13, we obtain the following theorem.

Theorem 2.2. Forn∈N∪ {0}, one has

1 −1ⁿ⁻¹ⁿ

l0

n

l En−lBl2−1ⁿnδ0,n−1. 2.14

In particular,

2n1

l0

2n1

l E2n1−lBl 0, forn∈N. 2.15

(6)

Let us consider the followingp-adic integral onZp for the product of Bernoulli and Euler polynomials as follows:

I3

Zp

BmxEnxdμx

^m

k0

n 0

m k

n

B_m−kE_n−

Zp

x^kxdμx

^m

k0

n 0

m k

n

Bm−kEn−Bk.

2.16

On the other hand, by1.10and1.13, one gets I3 −1^mn

Zp

Bm1−xEn1−xdμx

−1^mn^m

k0

n 0

m k

n

Bm−kEn−

Zp

1−x^kdμx

−1^mn^m

k0

n 0

m k

n

B_m−kE_n−kB_kδ_1,k −1^mnmB_m−11En1 −1^mnnBm1E_n−11

−1^mn^m

k0

n 0

m k

n

Bm−kEn−Bk −1^mnmBm−1EnnBmEn−1.

2.17

Equating2.16and2.17, one gets −1^mn11^m

k0

n 0

m k

n

B_m−kE_n−B_k −1^mnmB_m−1δ_1,m−12δ0,n−En

−1^mnnBmδ1,m2δ0,n−1−En−1 −1^mnnBmEn−1mBm−1En.

2.18

Forn∈N, by2.18, one gets −1^m11^m

k0

2n 0

m k

2n

Bm−kE2n−Bk

−1^m12nBmδ1,mE2n−1 −1^m2nBmE2n−1 −1^m12nδ1,mE2n−1.

2.19

Therefore, by2.19, one obtains the following theorem.

(7)

Theorem 2.3. Forn∈N, one has −1^m11^m

k0

2n 0

m k

2n

Bm−kE2n−Bk −1^m12nδ1,mE2n−1. 2.20 In particular, form∈N, one has

2m1

k0

2n 0

2m1 k

2n

B2m1−kE2n−Bk0. 2.21

By the same motivation, we consider the p-adic integral on Zp for the product of Bernoulli and Bernstein polynomials as follows:

I4

Zp

BmxBk,nxdμx wherem, n, k∈N∪ {0}. 2.22

From1.6and1.14, one gets

I4^m

0

m B_m−

Zp

xBk,nxdμx

n

k m 0

m Bm−

Zp

x^k1−x^n−kdμx

n

k m 0

n−k

j0

−1^j m

n−k

j Bm−Bkj.

2.23

On the other hand, I4 −1^m

Zp

Bm1−xB_n−k,n1−xdμx

−1^m n

k m 0

k j0

−1^j m

k

j B_m−

n−kjB_n−kjδ_1,n−kj

−1^m n

k n−kBm1δ0,k −1^m n

k mB_m−11δ0,k−−1^m n

k mBm1kδ_0,k−1

−1^m n

k m 0

k j0

−1^j m

k

j Bm−Bn−kj

−1^m n

k mBm−1−kBmδn,k −1^m n

k Bmδn,k1.

2.24

(8)

−1^m^m

0 n−k

j0

−1^j m

n−k

j Bm−Bkj

n−kBm1 mBm−11δ0,k−kBm1δ0,k−1 mBm−1−kBmδn,k

Bmδ_n,k1^m

0

k j0

−1^j m

k

j B_m−B_n−kj.

2.25

By2.25, we obtain the following theorem.

Theorem 2.4. Forn, m∈N, one has

2m 0

2n j0

−1^j 2m

2n

j B2m−Bj2nB2m^2m

0

2m

B2m−B2n. 2.26

Now, we consider the p-adic integral on Zp for the product of Euler and Bernstein polynomials as follows:

I5

Zp

EmxBk,nxdμx

^m

0

m Em−

Zp

xBk,nxdμx

n

k m 0

n−k

j0

−1^j m

n−k

j E_m−B_kj.

2.27

I5 −1^m

Zp

Bn−k,n1−xEm1−xdμx

−1^m n

k m 0

k j0

−1^j m

k j Em−

Zp

1−x^n−kjdμx

−1^m n

k m 0

k j0

−1^j m

k j

n−kjBn−kjδ1,n−kj Em−

(9)

−1^mn−k n

k Em1δ0,k −1^m n

k mEm−11δ0,k−−1^m n

k Em1kδ0,k−1

−1^m n

k m 0

k j0

−1^j m

k

j E_m−B_n−kj

−1^m n

k δ_n,k1Emδn,kmE_m−1−kEm.

2.28

−1^m^m

0 n−k

j0

−1^j m

n−k

j Em−Bkj

n−kEm1δ0,kmδ0,kEm−11−kEm1δ0,k−1

^m

0

k j0

−1^j m

k

j Em−Bn−kj

δn,k1Em mEm−1−kEmδn,k.

2.29

Theorem 2.5. Forn, m∈N, one has 2m

0

2n j0

−1^j 2m

2n

j E2m−Bj−2mE2m−1B2m2n. 2.30

Finally, we consider thep-adic integral onZpfor the product of Euler, Bernoulli, and Bernstein polynomials as follows:

I6

Zp

BrxEsxBk,nxdμx

n

k r 0

s j0

r

s

j Br−Es−j

Zp

x^kj1−x^n−kdμx

n

k r 0

s j0

n−k

i0

−1ⁱ r

s j

n−k

i Br−Es−jBkij.

2.31

On the other hand, by1.10,1.13, and1.14, one gets I6 −1^rs

Zp

Br1−xEs1−xBn−k,n1−xdμx

−1^rs n

k r 0

s j0

k i0

−1ⁱ r

s j

k

i Br−Es−j

Zp

1−x^n−kijdμx.

2.32

(10)

Equating2.31and2.32, we easily see that

−1^rs^r

0

s j0

n−k

i0

−1ⁱ r

s j

n−k

i Br−Es−jBkij

^r

0

s j0

k i0

−1ⁱ r

s j

k i

n−kijBn−kijδ1,n−kij

Br−Es−j

n−kBr1Es1δ0,krBr−11δ0,kEs1 sBr1Es−11δ0,k

−kBr1Es1δ0,k−1^r

0

s j0

k i0

−1ⁱ r

s j

k

i Br−Es−jBn−kij

δn,k1BrEs rBr−1EssBrEs−1−kBrEsδn,k.

2.33

Theorem 2.6. Forr, n, s∈N, one has 2r

0

2s j0

2n i0

−1ⁱ 2r

2s

j 2n

i B2r−E2s−jBij

−2sB2rE2s−1^r

0

2r

2l B2r−2lB2n2l2s−r s

j1

2s

2j−1 E2s−2j1B2n2r2j−2.

2.34

Acknowledgments

The authors express their sincere gratitude to the referees for their valuable suggestions and comments. This paper is supported in part by the Research Grant of Kwangwoon University in 2011.

References

1 T. Kim, “Symmetry p-adic invariant integral onZpfor Bernoulli and Euler polynomials,” Journal of Diﬀerence Equations and Applications, vol. 14, no. 12, pp. 1267–1277, 2008.

2 A. Bayad and T. Kim, “Identities involving values of Bernstein, q-Bernoulli, and q-Euler polynomials,”

Russian Journal of Mathematical Physics, vol. 18, no. 2, pp. 133–143, 2011.

3 A. Bayad, “Modular properties of elliptic Bernoulli and Euler functions,” Advanced Studies in Contemporary Mathematics, vol. 20, no. 3, pp. 389–401, 2010.

4 L. Jang, “A note on Kummer congruence for the Bernoulli numbers of higher order,” Proceedings of the Jangjeon Mathematical Society, vol. 5, no. 2, pp. 141–146, 2002.

5 L. C. Jang and H. K. Pak, “Non-Archimedean integration associated with q-Bernoulli numbers,”

Proceedings of the Jangjeon Mathematical Society, vol. 5, no. 2, pp. 125–129, 2002.

6 G. Kim, B. Kim, and J. Choi, “The DC algorithm for computing sums of powers of consecutive integers and Bernoulli numbers,” Advanced Studies in Contemporary Mathematics, vol. 17, no. 2, pp. 137–145, 2008.

(11)

7 T. Kim, “Some identities on the q-Euler polynomials of higher order and q-Stirling numbers by the fermionic p-adic integral onZp,” Russian Journal of Mathematical Physics, vol. 16, no. 4, pp. 484–491, 2009.

8 T. Kim, “Euler numbers and polynomials associated with zeta functions,” Abstract and Applied Analysis, vol. 2008, Article ID 581582, 11 pages, 2008.

9 T. Kim, “Note on the Euler numbers and polynomials,” Advanced Studies in Contemporary Mathematics, vol. 17, no. 2, pp. 131–136, 2008.

10 T. Kim, “On explicit formulas of p-adic q-L-functions,” Kyushu Journal of Mathematics, vol. 48, no. 1, pp. 73–86, 1994.

11 T. Kim, “Symmetry of power sum polynomials and multivariate fermionic p-adic invariant integral onZp,” Russian Journal of Mathematical Physics, vol. 16, no. 1, pp. 93–96, 2009.

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

13 T. Kim, “A note on q-Bernstein polynomials,” Russian Journal of Mathematical Physics, vol. 18, no. 2, pp.

14–50, 2011.

14 A. Kudo, “A congruence of generalized Bernoulli number for the character of the first kind,” Advanced Studies in Contemporary Mathematics, vol. 2, pp. 1–8, 2000.

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

16 S.-H. Rim, S. J. Lee, E. J. Moon, and J. H. Jin, “On the q-Genocchi numbers and polynomials associated with q-zeta function,” Proceedings of the Jangjeon Mathematical Society, vol. 12, no. 3, pp. 261–267, 2009.

17 C. S. Ryoo, “On the generalized Barnes type multiple q-Euler polynomials twisted by ramified roots of unity,” Proceedings of the Jangjeon Mathematical Society, vol. 13, no. 2, pp. 255–263, 2010.

18 I. Buyukyazici, “On Generalized q-Bernstein Polynomials,” Global Journal of Pure and Applied Mathematics, vol. 6, no. 3, pp. 1331–1348, 2010.

19 Y. Simsek, “Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions,” Advanced Studies in Contemporary Mathematics, vol. 16, no. 2, pp. 251–

278, 2008.

(12)

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}

Some New Identities on the Bernoulli and Euler Numbers

Research Article