Bernoulli Basis and the Product of Several Bernoulli Polynomials

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Volume 2012, Article ID 463659,12pages doi:10.1155/2012/463659

Research Article

Bernoulli Basis and the Product of

Several Bernoulli Polynomials

Dae San Kim

1

_{and Taekyun Kim}

2

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

2_{Division of General Education, Kwangwoon University, Seoul 139-701, Republic of Korea}

Correspondence should be addressed to Dae San Kim,dskim@sogang.ac.kr

Received 25 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.

We develop methods for computing the product of several Bernoulli and Euler polynomials by using Bernoulli basis for the vector space of polynomials of degree less than or equal ton.

1. Introduction

It is well known that, thenth Bernoulli and Euler numbers are defined by

n

l0

n l

Bl−Bnδ1,n, n

l0

n l

ElEn2δ0,n, 1.1

whereB0 E01 andδk,nis the Kronecker symbolsee1–20.

The Bernoulli and Euler polynomials are also defined by

Bnx n

l0

n l

Bn−lxl, Enx n

l0

n l

En−lxl. 1.2

Note that{B0x, B11, . . . , Bnx}forms a basis for the spacePn {px∈Qx|degpx≤

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So, for a givenpx∈Pn, we can write

px

n

k0

akBkx, 1.3

see8–12for uniquely determinedak∈Q.

Further,

ak

1

k!

pk−11−pk−10, wherepkx d

k_p_x

dxk ,

a0

1

0

ptdt, wherek1,2, . . . , n.

1.4

Probably,{1, x, . . . , xn}is the most natural basis for the spacePn. But{B0x, B1x, . . . , Bnx}

is also a good basis for the space Pn, for our purpose of arithmetical and combinatorial

applications.

What are common to Bnx, Enx, xn? A few proportion common to them are as

follows:

ithey are all monic polynomials of degreenwith rational coeﬃcients;

ii BnxnBn−1x,EnxnEn−1x,xnnxn−1;

iiiBnxdxBn1x/n1 c,

EnxdxEn1x/n1 c,

xn_dx_xn1_/_n

1 c.

In5,6, Carlitz introduced the identities of the product of several Bernoulli polynomials:

BmxBnx

∞

r0

m

2r

n

2r

m

B2rBmn−2rx

mn−2r −1

m1

× m!n!

mn!Bmn mn≥2,

1

0

BmxBnxBpxBqxdx −1mnpq

∞

r,s0

m

2r

n

2r

m ₂p

s

q

2s

p

×mn−2r−1!

pq−2s−1!

mnpq−2r−2s! BrBsBmnpq−2r−2

−1mp m!n!

mn!

p!q!

pq!BmnBpq.

1.5

In this paper, we will use1.4to derive the identities of the product of several Bernoulli and

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2. The Product of Several Bernoulli and Euler Polynomials

Let us consider the following polynomials of degreen:

px

i1···irj1···jsn

Bi1x· · ·BirxEj1x· · ·Ejsx, 2.1

where the sum runs over all nonnegative integersi1, . . . , ir,j1, . . . jssatisfyingi1· · ·irj1

· · ·jsn,rs1,r, s≥0.

Thus, from2.1, we have

pkx nrs−1nrs−2· · ·nrs−k

× ∞

i1···irj1···jsn−k

Bi1x· · ·BirxEj1x· · ·Ejsx.

2.2

Fork1,2, . . . , n, by1.4, we get

ak

1

k!

pk−11−pk−10

nkrs

nrs

i1···irj1···jsn−k1

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

nkrs

nrs

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

0≤a≤r

0≤c≤s

kr−n−1≤a≤r

r a

s c

−1c₂s−c

× ∞

i1···irj1···jsna1−k−r

Bi1· · ·BiaEj1· · ·Ejc

−

i1···irj1···jsn−k1

Bi1· · ·BirEj1· · ·Ejs

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭

.

2.3

From2.3, we note that

an

nrs n

nrs

i1···irj1···js1

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

nnrs

nrs −

1

21

r−

−1

2

s−

−1

2

rs

nnrs

nrsrs

nrs−1

n

(4)

an−1 1

nrs

nrs n−1

×

i1···irj1···js2

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

1

nrs

nrs n−1

1 6r 1 2 1 2

rs

2

−1

6r−

−1 2 −1 2

rs

2 0, a0 ₁ 0

ptdt

∞

i1···irj1···jsn

i1

l10

· · · ir

lr0

j1

p10

· · ·js

ps0

i1 l1 · · · ir lr j1 p1 · · · js ps

× Bi1−l1· · ·Bir−lrEj1−p1Ejs−ps

l1· · ·lrp1· · ·ps1.

2.4

Therefore, by1.3,2.1,2.3, and2.4, we obtain the following theorem.

Theorem 2.1. Forn∈Nwithn≥2, we have

i1···irj1···jsn

Bi1x· · ·BirxEj1x· · ·Ejsx

1

nrs

n−2

k1

nrs k × ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

0≤a≤r

0≤c≤s

kr−n−1≤a≤r

r a s c

−1c₂s−c

i1···iaj1···jcna1−k−r

Bi1· · ·BiaEj1· · ·Ejc

− ∞

i1···irj1···jsn−k1

Bi1· · ·BirEk1· · ·Ejs

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭

Bkx

nrs−1

n

Bnx

i1···irj1···jsn

i1

l10

. . .

ir

lr0

j1

p10

· · ·

js

ps0

i1 l1 · · · ir lr j1 p1 · · · js ps

l1· · ·lrp1· · ·ps1.

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Let us take the polynomialpxof degreenas follows:

px

i1···irj1···jstn

Bi1x· · ·BirxEj1x· · ·Ejsxx

t_,

2.6

Then, from2.6, we have

pkx nrsnrs−1· · ·nrs−k1

×

i1···irj1···jstn−k

t_, 2.7

By1.4and2.7, we get, fork1,2, . . . , n,

ak

1

k!

pk−11−pk−10

1

nrs1

k

×

i1···irj1···jstn−k1

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

t

_n_r_s₁

k

nrs1

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

0≤a≤r

0≤c≤s

kr−n−1≤a≤r

r a

s c

−1c₂s−c

×na1−k−r

t0

i1···iaj1···jct

Bi1· · ·BiaEj1· · ·Ejc

−

i1···irj1···jsn−k1

Bi1· · ·BirEj1· · ·Ejs

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭

,

2.8

Now, we look atanandan−1.

an

nrs1

n

nrs1

i1···irj1···jst1

t

nrs1

n

nrs1 1

2rs 1−

−1

2

rs

rs1

nrs1

n

nrs n

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an−1

_n_r_s₁

n−1

nrs1

i1···irj1···jst2

t

_n_r_s₁

n−1

nrs1 1

6r1

1 2 1 2

rs

2

1

2rs−

1

6r−

−1 2 −1 2

rs

2

1

nrs1

n−1

rs2

2

1 2

nrs n−1

,

2.9

From2.6, we note that

a0

₁

0

ptdt

i1

l10

· · ·ir

lr0

j1

p10

· · ·

js

ps0

i1 l1 · · · ir lr j1 p1 · · · js ps

×Bi1−l1· · ·Bir−lrEj1−p1Ejs−ps

1

l1· · ·lrp1· · ·pst1.

2.10

Therefore, by1.3,2.6,2.8,2.9, and2.10, we obtain the following theorem.

Theorem 2.2. Forn∈Nwithn≥2, one has

t

1

nrs1

n−2

k1

nrs1

k × ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

0≤a≤r

0≤c≤s

kr−n−1≤a≤r

r a s c

−1c₂s−cna1−k−r

t0

∞

i1···iaj1···jct

Bi1· · ·BiaEj1· · ·Ejc

−

i1···irj1···jsn−k1

Bi1· · ·BirEj1· · ·Ejs

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭

Bkx

1

2

nrs n−1

Bn−1x

nrs n

Bnx

i1

l10

· · ·ir

lr0

j1

p10

· · ·

js

ps0

i1 l1 · · · ir lr j1 p1 · · · js ps

×Bi1−l1· · ·Bir−lrEj1−p1· · ·Ejs−ps

× 1

l1· · ·lrp1· · ·pst1

.

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Consider the following polynomial of degreen:

px

∞

i1···irj1···jsn

1

i1!i2!· · ·ir!j1!· · ·js!Bi1x· · ·BirxEj1x· · ·Ejsx. 2.12

Then, from2.12, one has

pkx rsk

i1···irj1···jsn−k

Bi1x· · ·BirxEj1x· · ·Ejsx

i1!i2!· · ·ir!j1!· · ·js! . 2.13

By1.4and2.13, one gets, fork1,2, . . . , n,

ak

1

k!

pk−1₁₋_pk−1₀

rsk−1

k!

i1···irj1···jstn−k1

i1!i2!· · ·ir!j1!· · ·js!

rsk−1

k!

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

0≤a≤r

0≤c≤s

kr−n−1≤a≤r

r a

s c

−1c₂s−c

Bi1· · ·BiaEj1· · ·Ejc

i1!i2!· · ·ia!j1!· · ·jc!

−

i1···irj1···jsn−k1

1

i1!i2!· · ·ir!j1!· · ·js!Bi1· · ·BirEj1· · ·Ejs

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭

.

2.14

Now look atanandan−1:

an rs n−1

n!

i1···irj1···js1

i1!i2!· · ·ir!j1!· · ·js!

rsn−1

n! 1

2rs−

−1

2

rs

rsn

n! ,

an−1 rs

n−2

n−1!

i1···irj1···js2

i1!i2!· · ·ir!j1!· · ·js!

rsn−2

n−1!

1 2 1

6r

1 2 1 2

rs

2

− 1

2 1

6r−

1 2

−1

2

rs

2

0.

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It is easy to show that

a0

1

0

ptdt

i1···irj1···jsn

1

i1!· · ·ir!j1!· · ·js!

×i1

l10

· · ·ir

lr0

j1

p10

· · ·

js

ps0

Bi1−l1· · ·Bir−lrEj1−p1Ejs−ps

l1· · ·lrp1· · ·ps1

i1

l1

· · ·

ir

lr

j1

p1

· · ·

js

ps

.

2.16

Therefore, by1.3,2.14, and2.15, one obtains the following theorem.

i1···irj1···jsn

1

i1!i2!· · ·ir!j1!· · ·js!Bi1x· · ·BirxEj1x· · ·Ejsx

n−2

k1

rsk−1

k!

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

0≤a≤r

0≤c≤s

kr−n−1≤a≤r

r a

s c

−1c₂s−c

×

Bi1· · ·BiaEj1· · ·Ejc

i1!i2!· · ·ia!j1!· · ·jc!

−

i1···irj1···jsn−k1

1

i1!i2!· · ·ir!j1!· · ·js!

×Bi1· · ·BirEj1· · ·Ejs

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭

Bkx rs n

n! Bnx

i1···irj1···jsn

i1

l10

· · ·ir

lr0

j1

p10

· · ·

js

ps0

i1

l1

· · ·

ir

lr

j1

p1

· · ·

js

ps

i1!i2!· · ·ir!j1!· · ·js!

l1· · ·lrp1· · ·ps1

.

2.17

Take the polynomialpxof degreenas follows:

px

1

i1!i2!· · ·ir!j1!· · ·js!t!Bi1x· · ·BirxEj1x· · ·Ejsxx t_.

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Then, from2.18, one gets

pkx rs1k

×

i1···irj1···jstn−k

1

i1!i2!· · ·ir!j1!· · ·js!t!Bi1x· · ·BirxEj1x· · ·Ejsxx

t_. 2.19

By1.4and2.19, one gets, fork1, . . . , n,

ak 1

k!

pk−11−pk−10

rs1k−1

k!

i1···irj1···jstn−k1

1

i1!i2!· · ·ir!j1!· · ·js!t!

×Bi11· · ·Bir1Ej11· · ·Ejs1−Bi1· · ·BirEj1· · ·Ejs0

t

rs1k−1

k!

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

0≤a≤r

0≤c≤s

kr−n−1≤a≤r

r a

s c

t0

1

na1−k−r−t!

×

i1···iaj1···jct

1

i1!i2!· · ·ia!j1!· · ·jc!Bi1· · ·BiaEj1· · ·Ejc

−

i1···irj1···jsn−k1

Bi1· · ·BirEj1· · ·Ejs

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭

.

2.20

Now look atanandan−1:

an rs1 n−1

n!

i1···irj1···jst1

1

i1!i2!· · ·ir!j1!· · ·js!t!

t

rs1n−1

n!

1

2rs 1−

−1

2

rs

rs1n−1

n! rs1

rs1n

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an−1 rs1

n−2

n−1!

i1···irj1···jst2

1

i1!i2!· · ·ir!j1!· · ·js!t!

t

r s1n−2

n−1!

1 2 1

6r

1

2

1 2 1 2

rs

2

1

2rs−

1 2 1

6r−

−1

2

−1

2

rs

2

rs1n−2

n−1!

rs1 2

rs1n−1

2n−1! .

2.21

From2.18, one can derive the following identity:

a0

1

0

ptdt

1

i1!· · ·ir!j1!· · ·js!t!

₁

0

t_dt

1

i1!· · ·ir!j1!· · ·js!t! i1

l10

· · ·ir

lr0

j1

p10

· · ·

×

js

ps0

i1

l1

· · ·

ir

lr

j1

p1

· · ·

js

ps

Bi1−l1· · ·Bir−lrEj1−p1Ejs−ps

1

l1· · ·lrp1· · ·pst1.

2.22

Therefore, by1.3,2.20,2.21, and2.22, one obtains the following theorem.

1

i1!i2!· · ·ir!j1!· · ·js!t!Bi1x· · ·BirxEj1x· · ·Ejsxx t

n−2

k1

rs1k−1

k!

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

0≤a≤r

0≤c≤s

kr−n−1≤a≤r

r a

s c

t0

1

na1−k−r−t!

×

i1···iaj1···jct

1

i1!i2!· · ·ia!j1!· · ·jc!Bi1· · ·BiaEj1· · ·Ejc

−

i1···irj1···jsn−k1

Bi1· · ·BirEj1· · ·Ejs

i1!i2!· · ·ir!j1!· · ·js!

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭

(11)

r₂s1n−1

n−1! Bn−1x

rs1n

n!

i1

l10

· · ·ir

lr0

j1

p10

· · ·

js

ps0

×

i1

l1

· · ·ir

lr

j1

p1

· · ·js

ps

i1!· · ·ir!j1!· · ·js!t! Bi1−l1· · ·Bir−lrEj1−p1Ejs−ps

1

l1· · ·lrp1· · ·pst1.

2.23

Acknowledgments

This research was supported by Basic Science Research Program through the National

Research Foundation of Korea NRF funded by the Ministry of Education, Science and

Technology 2012R1A1A2003786.

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Bernoulli Basis and the Product of Several Bernoulli Polynomials

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