Euler Basis, Identities, and Their Applications

(1)

Volume 2012, Article ID 343981,15pages doi:10.1155/2012/343981

Research Article

Euler Basis, Identities, and Their Applications

D. S. Kim

1

_{and T. Kim}

2

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

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

Correspondence should be addressed to D. S. Kim,[email protected]

Received 11 June 2012; Accepted 9 August 2012

Academic Editor: Yilmaz Simsek

Copyrightq2012 D. S. Kim and T. Kim. 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.

LetVn {px∈Qx|degpx≤n}be then1-dimensional vector space overQ. We show

that{E0x, E1x, . . . , Enx}is a good basis for the spaceVn, for our purpose of arithmetical and

combinatorial applications. Thus, ifpx ∈ Qxis of degree n, thenpx nl0blElx for

some uniquely determinedbl ∈Q. In this paper we develop method for computingblfrom the

information ofpx.

1. Introduction

The Euler polynomials,Enx, are given by

2

et₁e

xt_eExt∞

n0

Enxtn

n!, 1.1

see1–20with the usual convention about replacingEn_x_by_Enx_{. In the special case,} x 0, En0 Enare called thenth Euler numbers. The Bernoulli numbers are also defined by

t

et₋₁ e

Bt∞

n0

Bnt

n

n!, 1.2

see 1–20 with the usual convention about replacing Bn _by _Bn_{. As is well known, the}

Bernoulli polynomials are given by

Bnx n

l0

n l

Blxn−l

n

l0

n l

(2)

see9–15From1.1,1.2, and1.3, we note that

Bn1−Bnδ1,n, En1 En2δ0,n, 1.4

whereδk,nis the kronecker symbol.

Letm, n∈Zwithmn≥2. The formula

BmxBnx

r

m

2r

n

2r

m

B2rBmn−2rx

mn−2r −1

m1m!n!Bmn

mn! , 1.5

is proved in 4–6. Let Vn {px ∈ Qx | degpx ≤ n} be the n1-dimensional vector space over Q. Probably, {1, x, . . . , xn_} _{is the most natural basis for this space. But}

{E0x, E1x, . . . , Enx}is also a good basis for the spaceVn, for our purpose of arithmetical

and combinatorial applications. Thus, ifpx∈Qxis of degreen, then

px n

l0

blElx, 1.6

for some uniquely determinedbl∈Q. Further,

bk 1

2k!

pk1 pk0 k0,1,2, . . . , n, 1.7

where pkx dk_px/dxk_{. In this paper we develop methods for computing} _bl

from the information of px. Apply these results to arithmetically and combinatorially interesting identities involvingE0x, E1x, . . . , Enx, B0x, . . . , Bnx. Finally, we give some applications of those obtained identities.

2. Euler Basis, Identities, and Their Applications

Let us takepxthe polynomial of degreenas follows:

px n

k0

BkxBn−kx. 2.1

From2.1, we have

pkx _nn₋1!

k1!

n

lk

(3)

By1.7and2.2, we get

bk 1

2k!

pk1 pk0

1

2

n1

k

n

lk

{Bl−kδ1,l−kBn−lδ1,n−l Bl−kBn−l},

2.3

Thus, we have

bk

n1

k

n

lk

Bl−kBn−lBn−k−1

, 0≤k≤n−3, 2.4

bn−2 7

72n

n2₋₁_, _bn_n₁_, _bn−1₀_. _2.5

By1.6,2.1,2.3, and2.4, we get

n

k0

BkxBn−kx

n−3

k0

n1

k

n

lk

Bl−kBn−lBn−k−1

Ekx 7

72n

n2−1En−2x n1Enx.

2.6

Let us consider the following triple identities:

px

rstnBrxBsxBtx

n

k0

bkEkx, 2.7

where the sum runs over allr, s, t∈Zwithrstn. Thus, by2.7, we get

pkx n2n1nn−1· · ·n−k3

rstn−k

BrxBsxBtx. _2.8

From1.7and2.8, we have

bk 1

2k!

pk1 pk0

_n2

k

2

rstn−k

{Br1Bs1Bt1 BrBsBt}

_n2

k

2

rstn−k

BrBsBt

rstn−k

δ1,rBsBt

rstn−k

(4)

rstn−k

BrBsδ1,t

rstn−k

δ1,rδ1,sBt

rstn−k

δ1,rBsδ1,t

rstn−k

Brδ1,sδ1,t

rstn−k

δ1,rδ1,sδ1,t

.

2.9

Therefore, by2.7and2.9, we obtain the following theorem.

Theorem 2.1. Forr, s, t∈Z, andn∈Nwithn≥3, one has

rstn

BrxBsxBtx

1

2

n−2

k0

n2

k

2

rstn−k

BrBsBt3

rsn−k−1

BrBs3Bn−k−2δk,n−3

Ekx

n2

2

Enx.

2.10

Let us take the polynomialpxas follows:

px

rstn

BrxBsxEtx. _2.11

Then, by2.11, we get

rstn−k

BrxBsxEtx. _2.12

From1.6,1.7, and2.12, we have

bk 1

2k!

pk1 pk0

_n2

k

2

rstn−k

{Br1Bs1Et1 BrBsEt}

_n2

k

2

rstn−k

{Br δ1,rBsδ1,s−Et2δ0,t BrBsEt}

_n2

k

2

−

rstn−k

δ1,rBsEt−

rstn−k

Brδ1,sEt2

rstn−k BrBsδ0,t

−

rstn−k

δ1,rδ1,sEt2

rstn−k

δ1,rBsδ0,t2

rstn−k

Brδ1,sδ0,t

2

rstn−k

δ1,rδ1,sδ0,t

.

(5)

Note that

bn−1

n2

n−1

−

st0

BsEt−

rt0

BrEt2

rs1

BrBs−02B02B02·0

1

2

n2

n−1

{−1−12B1B1 22}0.

2.14

Therefore, we obtain the following theorem.

Theorem 2.2. Forn∈Nwithn≥2, one has

rstn

BrxBsxEtx

1

2

n−2

k0

n2

k

2

rsn−k

BrBs−2

rtn−k−1

BrEt−En−k−24Bn−k−12δk,n−2

Ekx

n2

2

Enx.

2.15

Remark 2.3. By the same method, we obtain the following identities.

I

rstn

BrxEsxEtx

1

2

n−2

k0

n2

k

2

rstn−k

BrEsEt

stn−k−1

EsEt−4

rsn−k

BrEs4Bn−k−4En−k−1

Ekx

n2

2

Enx.

2.16

II

rstn

ErxEsxEtx

3

n−2

k0

n2

k _rsn−kErEs−2En−k

Ekx

n2

2

Enx.

2.17

Let us consider the polynomialpxas follows:

px

rstn

BrxBsxxt_.

(6)

Thus, by2.18, we get

rstn−k

BrxBsxxt. _2.19

From1.6,1.7,2.18, and2.19, we have

bk 1

2k!

pk1 pk0

_n2

k

2

rstn−k

Br1Bs1 BrBs0t

_n2

k

2

rstn−k

Brδ1,rBsδ1,s BrBs0t

_n2

k

2

rstn−k

BrBs

rstn−k

Brδ1,s

rstn−k

δ1,rBs

rstn−k

δ1,rδ1,s

rstn−k

BrBs0t

.

2.20

Here we note that

rstn−k

BrBs

n−k

t0

rsn−k−t

BrBs

n−k

t0

rst

BrBs

rstn−k

Brδ1,s

⎧ ⎪ ⎨ ⎪ ⎩

n−k−1

r0 Br, if k≤n−1,

0, if kn,

rstn−k

Bsδ1,r

⎧ ⎪ ⎨ ⎪ ⎩

n−k−1

r0 Br, if k≤n−1,

0, if kn,

rstn−k

δ1,rδ1,s

1, ifk≤n−2,

0, ifkn−1 orn,

rstn−k

BrBs0t

rsn−k

BrBs, ∀k.

2.21

It is easy to show that

bn−1 1

2

n2

n−1 _rs0BrBs2

rs1

BrBs2B0

1

2

n2

n−1

{12B1B2 2} 1

2

n2

n−1

.

2.22

(7)

rstn

BrxBsxxt

1

2

n−2

k0

n2

k

n−k−1

t0

rst

BrBs2

rsn−k

BrBs2

n−k−1

r0

Br 1

Ekx

1

2

n2

n−1

En−1x

n2

n

Enx.

2.23

Remark 2.5. By the same method, we can derive the following identities.

I

rstnBrxEsxx

t

1

2

n−2

k0

n2

k

−n−k−1

t0

rst

BrEs−

n−k−1

s0

Es2

n−k

r0

Br 2

Ekx

1

2

n2

n−1

En−1x

n2

n

Enx.

2.24

II

rstn

ErxEsxxt

1

2

n−2

k0

n2

k

_n−k−1

t0

rst

ErEs2

rsn−k

ErEs−4

n−k

r0

Er4

Ekx

1

2

n2

n−1

En−1x

n2

2

Enx.

2.25

Now we generalize the above consideration to the completely arbitrary case. Let

px

i1···irj1···jsn

Bi1x· · ·BirxEj1x· · ·Ejsx, 2.26

where the sum runs over all nonnegative integersi1, i2, . . . , ir, j1, . . . , jssatisfyingi1i2· · ·

irj1· · ·jsn. From2.26, we note that

pkx nrs−1· · ·nrs−k

i1···irj1···jsn−k

Bi1x· · ·Birx×Ej1x· · ·Ejsx.

(8)

By1.6,1.7,2.18, and2.27, we get

bk 1

2k!

pk1 pk0

1

2

nrs−1

k

Bi11· · ·Bir1Ej11· · ·Ejs1 Bi1· · ·BirEj1· · ·Ejs

1

2

nrs−1

k

×

{Bi1δ1,i1· · ·Birδ1,ir

×−Ej12δ0,j1

· · ·−Ej_s2δ0,js

Bi1· · ·BirEj1· · ·Ejs

1

2

nrs−1

k

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

0≤a≤r 0≤c≤s a≥kr−n

r a

s c

−1c2s−c ×

i1···iaj1···jcna−k−r

Bi1· · ·BiaEj1· · ·Ejc

Bi1· · ·BirEj1· · ·Ejs

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ .

2.28

Note that

bn 1

2

nrs−1

n

⎧_⎨

⎩

0≤c≤s

s c

−1c2s−c×

i1···irj1···jc0

Bi1· · ·BirEj1· · ·Ejc

i1···irj1···js0

Bi1· · ·BirEj1· · ·Ejs

⎫ ⎬ ⎭

1

2

nrs−1

n

2−1s1

nrs−1

n

,

bn−1 1

2

nrs−1

n−1

⎧⎪_⎨

⎪ ⎩

r−1≤a≤r 0≤c≤s

r a

s c

−1c2s−c

×

i1···iaj1···jc1a−r

Bi1· · ·BiaEj1· · ·Ejc

Bi1· · ·BirEj1· · ·Ejs

(9)

1

2

nrs−1

n−1

r2−1s

0≤c≤s

s c

−1c2s−c

−1

2rc

−1

2rs

1

2

nrs−1

n−1

r−1

2r 1 2s−

1 2r−

1 2s

0.

2.29

Therefore, by1.6,2.28, and2.29, we obtain the following theorem.

Bi1x· · ·BirxEj1x· · ·Ejsx

1

2

n−2

k0

nrs−1

k

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

r a

s c

−1c2s−c

×

Bi1· · ·BiaEj1· · ·Ejc

Bi1· · ·BirEj1· · ·Ejs

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭

Ekx

nrs−1

n

Enx.

2.30

Let us consider the polynomialpxof degreenas

px

ti1···irj1···jsn

Bi1x· · ·BirxEj1x· · ·Ejsxx

t_.

2.31

Then, from2.31, we have

pkx nrsnrs−1· · ·nrs−k1

×

i1···irj1···jstn−k

(10)

By1.7and2.32, we get

bk 1

2k!

pk1 pk0

1

2

nrs

k

Bi11· · ·Bir1Ej11· · ·Ejs1 Bi1· · ·BirEj1· · ·Ejs0

t

1

2

nrs

k

×

{Bi1δ1,i1· · ·Bir δ1,ir

×−Ej02δ0,j1

· · ·−Ejs2δ1,js

Bi1· · ·BirEj1· · ·Ejs0

t

2.33

From2.33, we can derive the following equation:

bk 1

2

nrs

k

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

r a

s c

−1c2s−c×

na−k−r

t0

i1···iaj1···jct

Bi1· · ·BiaEj1· · ·Ejc

Bi1· · ·BirEj1· · ·Ejs

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ .

2.34

Observe now that

bn 1

2

nrs

n

⎧_⎨

⎩ s

c0

s c

−1c2s−c×

Bi1· · ·BirEj1· · ·Ejc

Bi1· · ·BirEj1· · ·Ejs

⎫ ⎬ ⎭

1

2

nrs

n

2−1s1

nrs

n

,

(11)

bn−1 1

2

nrs

n−1

⎧⎪_⎨

⎪ ⎩

r−1≤a≤r 0≤c≤s

r a

s c

−1c2s−c×

1a−r

t0

Bi1· · ·BiaEj1· · ·Ejc

Bi1· · ·BirEj1· · ·Ejs

⎫ ⎪ ⎬ ⎪ ⎭

1

2

nrs

n−1

r1−1

2r 1 2s−

1 2r−

1 2s

1

2

nrs

n−1

.

2.36

Therefore, by1.6,2.31,2.34,2.35, and2.36, we obtain the following theorem.

i1···irj1···jstn

t

1

2

n−2

k0

nrs

k

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

r a

s c

−1c2s−c

×na−k−r

t0

Bi1· · ·BiaEj1· · ·Ejc

Bi1· · ·BirEj1· · ·Ejs

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭

Ekx

1

2

nrs

n−1

En−1x

nrs

n

Enx.

2.37

Let us consider the following polynomial of degreen.

px

1

i1!· · ·ir!j1!· · ·js!Bi1x· · ·BirxEj1x· · ·Ejsx. 2.38

pkx rsk

1

(12)

From1.7, we have

bk 1

2k!

pk1 pk0

rsk

2k!

1

i1!· · ·ir!j1!· · ·js!

×Bi11· · ·Bir1×Ej11· · ·Ejs1 Bi1· · ·BirEj1· · ·Ejs

rsk

2k!

1

i1!· · ·ir!j1!· · ·js!

×Bi1δ1,i1· · ·Birδ1,ir×

−Ej12δ0,j1

· · ·−Ejs2δ0,js

Bi1· · ·BirEj1· · ·Ejs

.

2.40

bk rs

k

2k!

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

r a

s c

−1c2s−c×

Bi1· · ·BiaEj1· · ·Ejc

i1!· · ·ia!j1!· · ·jc!

Bi1· · ·BirEj1· · ·Ejs

i1!· · ·ir!j1!· · ·js!

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ .

2.41

Now, we note that

bn rs

n

2n!

⎧ ⎨ ⎩ s

c0

s

c

−1c2s−c

×

Bi1· · ·BirEj1· · ·Ejc

i1!· · ·ir!j1!· · ·jc!

Bi1· · ·BirEj1· · ·Ejs

i1!· · ·ir!j1!· · ·js!

⎫ ⎬ ⎭

rsn

2n!

2−1s1 rs

n

(13)

bn−1 rs n−1

2n−1!

⎧ ⎪ ⎨ ⎪ ⎩ r−1≤a≤r 0≤c≤s r a s c

−1c2s−c

×

i1···iaj1···jc1a−r

Bi1· · ·BiaEj1· · ·Ejc

i1!· · ·ia!j1!· · ·jc!

Bi1· · ·BirEj1· · ·Ejs

i1!· · ·ir!j1!· · ·js!

⎫ ⎪ ⎬ ⎪ ⎭

rsn−1

2n−1!

r2−1s

s c0 s c

−1c2s−c

−1

2rc

−1

2rs

0.

2.42

Therefore, by1.6,2.38,2.41, and2.42, we obtain the following theorem.

Bi1x· · ·BirxEj1x· · ·Ejsx

i1!· · ·ir!j1!· · ·js!

1

2

n−2

k0

rsk

k! ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 0≤a≤r 0≤c≤s a≥kr−n r a s c

−1c2s−c×

Bi1· · ·BiaEj1· · ·Ejc

i1!· · ·ia!j1!· · ·jc!

Bi1· · ·BirEj1· · ·Ejs

i1!· · ·ir!j1!· · ·js!

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ Ekx

rsn

n! Enx.

2.43

By the same method, we can obtain the following identity:

i1···irj1···jstn

Bi1x· · ·BirxEj1x· · ·Ejsxxt

i1!· · ·ir!j1!· · ·js!t!

1

2

n−2

k0

rs1k

k! ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 0≤a≤r 0≤c≤s a≥kr−n r a s c

(14)

×na−k−r t0

1

na−k−r−t!

Bi1· · ·BiaEj1· · ·Ejc

i1!· · ·ia!j1!· · ·jc!

Bi1· · ·BirEj1· · ·Ejs

i1!· · ·ir!j1!· · ·js!

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭

Ekx

rs1n−1

2n−1! En−1x

rs1n

n! Enx.

2.44

Acknowledgments

This paper was supported by Basic Science Research Program through the National Research Foundation of Korea NRFfunded by the Ministry of Education, Science and Technology 2012R1A1A2003786.

References

1 S. Araci, D. Erdal, and J. J. Seo, “A study on the fermionicp-adicq-integral representation onZp

associated with weightedq-Bernstein andq-Genocchi polynomials,” Abstract and Applied Analysis, vol. 2011, Article ID 649248, 10 pages, 2011.

2 A. Bayad, “Modular properties of elliptic Bernoulli and Euler functions,” Advanced Studies in

Contemporary Mathematics, vol. 20, no. 3, pp. 389–401, 2010.

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

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

4 L. Carlitz, “Note on the integral of the product of several Bernoulli polynomials,” Journal of the London

Mathematical Society, vol. 34, pp. 361–363, 1959.

5 L. Carlitz, “Multiplication formulas for products of Bernoulli and Euler polynomials,” Pacific Journal

of Mathematics, vol. 9, pp. 661–666, 1959.

6 L. Carlitz, “Arithmetic properties of generalized Bernoulli numbers,” Journal f ¨ur die Reine und

Angewandte Mathematik, vol. 202, pp. 174–182, 1959.

7 N. S. Jung, H. Y. Lee, and C. S. Ryoo, “Some relations between twistedh, q-Euler numbers with weightαandq-Bernstein polynomials with weightα,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 176296, 11 pages, 2011.

8 D. S. Kim, “Identities of symmetry forq-Euler polynomials,” Open Journal of Discrete Mathematics, vol. 1, no. 1, pp. 22–31, 2011.

9 D. S. Kim, “Identities of symmetry for generalized Euler polynomials,” International Journal of

Combinatorics, vol. 2011, Article ID 432738, 12 pages, 2011.

10 T. Kim, “On the weightedq-Bernoulli numbers and polynomials,” Advanced Studies in Contemporary

Mathematics, vol. 21, no. 2, pp. 207–215, 2011.

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

12 T. Kim, “Some identities on theq-Euler polynomials of higher order andq-Stirling numbers by the fermionicp-adic integral onZp,” Russian Journal of Mathematical Physics, vol. 16, no. 4, pp. 484–491,

2009.

13 B. Kurt and Y. Simsek, “Notes on generalization of the Bernoulli type polynomials,” Applied

Mathematics and Computation, vol. 218, no. 3, pp. 906–911, 2011.

(15)

15 H. Ozden, “p-adic distribution of the unification of the Bernoulli, Euler and Genocchi polynomials,”

Applied Mathematics and Computation, vol. 218, no. 3, pp. 970–973, 2011.

16 H. Ozden, I. N. Cangul, and Y. Simsek, “On the behavior of two variable twisted p-adic Euler q-l-functions,” Nonlinear Analysis, vol. 71, no. 12, pp. e942–e951, 2009.

17 S.-H. Rim, A. Bayad, E.-J. Moon, J.-H. Jin, and S.-J. Lee, “A new construction on theq-Bernoulli polynomials,” Advances in Diﬀerence Equations, vol. 2011, article 34, 2011.

18 C. S. Ryoo, “Some relations between twistedq-Euler numbers and Bernstein polynomials,” Advanced

Studies in Contemporary Mathematics, vol. 21, no. 2, pp. 217–223, 2011.

19 Y. Simsek, “Complete sum of products ofh, q-extension of Euler polynomials and numbers,” Journal

of Diﬀerence Equations and Applications, vol. 16, no. 11, pp. 1331–1348, 2010.

(16)

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 MathematicsJournal of

http://www.hindawi.com Volume 2014 Mathematical PhysicsAdvances in

Complex Analysis

Journal of

Optimization

Journal of

Combinatorics

International Journal of

Journal of

Function Spaces

Abstract and Applied Analysis Hindawi Publishing Corporation

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporation http://www.hindawi.com Volume 2014

The Scientific

World Journal

Discrete Dynamics in Nature and Society Hindawi Publishing Corporation

Discrete Mathematics

Journal of