Sums of finite products of Bernoulli functions

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R E S E A R C H

Open Access

Sums of ﬁnite products of Bernoulli

functions

Ravi P Agarwal

1

_{, Dae San Kim}

2

_{, Taekyun Kim}

3,4*

_{and Jongkyum Kwon}

5

*_{Correspondence: tkkim@kw.ac.kr} 3_{Department of Mathematics,} Kwangwoon University, Seoul, 139-701, Republic of Korea 4_{Department of Mathematics,} College of Science, Tianjin Polytechnic University, Tianjin, 300160, China

Full list of author information is available at the end of the article

Abstract

In this paper, we consider three types of functions given by sums of ﬁnite products of Bernoulli functions and derive their Fourier series expansions. In addition, we express each of them in terms of Bernoulli functions.

MSC: 11B68; 42A16

Keywords: Fourier series; sums of ﬁnite products of Bernoulli functions

1 Introduction

As is well known, theBernoulli polynomials Bm(x) are given by the generating function

t et_{– }e

xt₌

∞

m= Bm(x)

tm

m! (see [–]). (.)

Whenx= ,Bm=Bm() are called Bernoulli numbers. For any real numberx, we let

x=x– [x]∈[, ) (.)

denote the fractional part ofx.

Fourier series expansion of higher-order Bernoulli functions was treated in the recent paper []. Here we will consider the following three types of functions given by sums of ﬁnite products of Bernoulli functions and derive their Fourier series expansions. In addition, we will express each of them in terms of Bernoulli functions.

() αm(x) =c+c+···+cr=m,c,...,cr≥Bc(x)Bc(x)· · ·Bcr(x)(m≥);

() βm(x)=

c+c+···+cr=m,c,...,cr≥



c!c!···cr!Bc(x)Bc(x)· · ·Bcr(x)(m≥);

() γr,m(x) =c+c+···+cr=m,c,...,cr≥



cc···crBc(x)Bc(x)· · ·Bcr(x)(m≥r).

For elementary facts about Fourier analysis, the reader may refer to any book (for example, see [, ]).

As toβm(x), we note that the next polynomial identity follows immediately from

The-orems . and ., which is in turn derived from the Fourier series expansion ofβm(x):

c+c+···+cr=m

 c!c!· · ·cr!

Bc(x)Bc(x)· · ·Bcr(x) =

 rm++

m

j= rj–

j! m–j+Bj(x),

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where

l=

max{,r–l}≤a≤r–

r a

c+c+···+ca=l+a–r

BcBc· · ·Bca

c!c!· · ·ca!

. (.)

The obvious polynomial identities can be derived also forαm(x) andγm(x) from

The-orems . and ., and TheThe-orems . and ., respectively. It is remarkable that from the Fourier series expansion of the function m_k_=–_k₍_m_–_k₎Bk(x)Bm–k(x) we can derive the

Faber-Pandharipande-Zagier identity (see [–]) and the Miki identity (see [–]).

2 The function

α

m(x)

Letαm(x) =

c+c+···+cr=mBc(x)Bc(x)· · ·Bcr(x) (m≥). Here the sum runs over all

non-negative integersc,c, . . . ,crwithc+c+· · ·+cr=m(r≥). Then we will consider the

function

αm

x=

c+c+···+cr=m

Bc

xBc

x· · ·Bcr

x, (.)

deﬁned on (–∞,∞), which is periodic with period . The Fourier series ofαm(x) is

∞

n=–∞

A(_nm)eπinx, (.)

where

A(_nm)=



 αm

xe–πinxdx

=





αm(x)e–πinxdx. (.)

Before proceeding further, we need to observe the following.

α_m(x) =

c+c+···+cr=m

cBc–(x)Bc(x)· · ·Bcr(x)

+· · ·+crBc(x)Bc(x)· · ·Bcr–(x)Bcr–(x)

=

c+c+···+cr=m,c≥

cBc–(x)Bc(x)· · ·Bcr(x)

+· · ·+

c+c+···+cr=m,cr≥

crBc–(x)Bc(x)· · ·Bcr(x)

= (m+r– )

c+c+···+cr=m–

Bc(x)Bc(x)· · ·Bcr(x)

= (m+r– )αm–(x). (.)

From this, we have

αm+(x) m+r

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and





αm(x)dx=

 m+r

αm+() –αm+()

. (.)

Form≥, we put

m=αm() –αm()

=

c+c+···+cr=m

Bc()Bc()· · ·Bcr() –BcBc· · ·Bcr

=

c+c+···+cr=m

(Bc+δ,c)· · ·(Bcr+δ,cr) –BcBc· · ·Bcr

=

≤a≤r a≥r–m

r a

c+c+···+ca=m+a–r

BcBc· · ·Bca–

c+c+···+cr=m

BcBc· · ·Bcr

=

max{,r–m}≤a≤r–

r a

c+c+···+ca=m+a–r

BcBc· · ·Bca, (.)

where we understand that, forr–m≤ anda= , the inner sum isδm,r.

Observe here that the sum over allc+c+· · ·+cr=mof any term withaofBceandb

ofδ,cf (≤e,f≤r,a+b=r), all give the same sum

c+c+···+cr=m

Bc· · ·Bcaδ,ca+· · ·δ,ca+b

=

c+c+···+ca=m+a–r

BcBc· · ·Bca, (.)

which is not an empty sum as long asm+a–r≥, i.e.,a≥r–m. Thus

αm() =αm() ⇐⇒ m=  (.)

and





αm(x)dx=



m+rm+. (.)

Now, we are ready to determine the Fourier coeﬃcientsA(nm).

Case :n= .

A(_nm)=





αm(x)e–πinxdx

= – 

πin αm(x)e –πinx

+  πin





α_m(x)e–πinxdx

=m+r–  πin A

(m–)

n –

 πinm

=m+r–  πin

m+r–  πin A

(m–)

n –

 πinm–

– 

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=(m+r– )

Let us recall the following facts about Bernoulli functionsBm(x):

(a) form≥,

integersmwithm=  and discontinuous with jump discontinuities at integers for those

positive integersmwithm= .

Assume ﬁrst thatmis a positive integer withm= . Thenαm() =αm(). Henceαm(x)

is piecewiseC∞and continuous. Thus the Fourier series ofαm(x) converges uniformly

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We can now state our ﬁrst result.

Theorem . For each positive integer l,we let

l=

Assume thatm= for a positive integer m.Then we have the following.

(a) _c_₊_c_₊_···₊_c_r₌_mBc(x)Bc(x)· · ·Bcr(x)has the Fourier series expansion

for allx∈R,where the convergence is uniform. (b)

piecewiseC∞and discontinuous with jump discontinuities at integers. The Fourier series ofαm(x) converges pointwise toαm(x) forx∈/Zand converges to

Now, we can state our second result.

Theorem . For each positive integer l,we let

l=

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(b) 

deﬁned on (–∞,∞), which is periodic with period . The Fourier series ofβm(x) is

∞

Before proceeding further, we need to observe the following.

β_m(x) =

From this, we have

βm+(x) r

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and





βm(x)dx=

 r

βm+() –βm+()

. (.)

Let

m=βm() –βm()

=

c+c+···+cr=m

Bc()Bc()· · ·Bcr()

c!c!· · ·cr!

–

c+c+···+cr=m

c!c!· · ·cr!

=

c+c+···+cr=m

(Bc+δ,c)(Bc+δ,c)· · ·(Bcr+δ,cr)

c!c!· · ·cr!

–

c+c+···+cr=m

c!c!· · ·cr!

=

max{,r–m}≤a≤r–

r a

c+c+···+ca=m+a–r

c!c!· · ·ca!

, (.)

where we understand that, forr–m≤ anda= , the inner sum isδm,r.

Observe here that the sum over allc+c+· · ·+cr=mof any term withaofBceandb

ofδ,cf (≤e,f≤r,a+b=r), all give the same sum

c+c+···+cr=m

Bc· · ·Bcaδ,ca+· · ·δ,ca+b

c!c!· · ·cr!

=

c+c+···+ca=m+a–r

c!c!· · ·cr!

, (.)

which is not an empty sum as long asm+a–r≥, i.e.,a≥r–m. Also, we have

βm() =βm() ⇔ m=  (.)

and





βm(x)dx=



rm+. (.)

Now, we would like to determine the Fourier coeﬃcientsB(nm).

Case :n= .

B(m)

n =





βm(x)e–πinxdx

= – 

πin βm(x)e –πinx

+  πin





β_m(x)e–πinxdx

= –  πin

βm() –βm()

+ r

πin





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= r

integersmwithm=  and discontinuous with jump discontinuities at integers for those

positive integersmwithm= .

Assume ﬁrst thatm=  for a positive integerm. Thenβm() =βm(). Henceβm(x) is

piecewiseC∞and continuous. Thus the Fourier series ofβm(x) converges uniformly to

βm(x), and

Now, we can state our ﬁrst result.

Theorem . For each positive integer l,we let

l=

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(a) _c_₊_c_₊_···₊_c_r₌_m_c 

is piecewiseC∞and discontinuous with jump discontinuities at integers. Thus the Fourier series ofβm(x) converges pointwise toβm(x) forx∈/Zand converges to

Theorem . For each positive integer l,let

l=

(a) 

Here the convergence is pointwise.

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 rm++

m

j= rj–

j! m–j+Bj

x

=

c+c+···+cr=m

 c!c!· · ·cr!

BcBc· · ·Bcr+



m for x∈Z.

Here Bj(x)is the Bernoulli function.

4 The function

γ

r,m(x)

Letγr,m(x) =

c+c+···+cr=m,c,...,cr≥cc···crBc(x)Bc(x)· · ·Bcr(x) (m≥r≥). Here the sum

is over all positive integersc,c, . . . ,crwithc+c+· · ·+cr=m.

γ_r_,_m(x) =

c+c+···+cr=m,c,...,cr≥

 c· · ·cr

Bc–(x)Bc(x)· · ·Bcr(x)

+

c+c+···+cr=m,c,...,cr≥

 cc· · ·cr

Bc(x)Bc–(x)· · ·Bcr(x)

+· · ·+

c+c+···+cr=m,c,...,cr≥  cc· · ·cr–

Bc(x)Bc(x)· · ·Bcr–(x)

=

c+···+cr=m–,c,...,cr≥  c· · ·cr

Bc(x)· · ·Bcr(x)

+

c+···+cr=m–,c,...,cr≥  c· · ·cr

Bc(x)· · ·Bcr(x)

+· · ·+

c+c+···+cr–=m–,c,...,cr–≥

 cc· · ·cr–

Bc(x)· · ·Bcr–(x)

+

c+c+···+cr=m–,c,...,cr≥

 cc· · ·cr–

Bc(x)· · ·Bcr(x)

=rγr–,m–(x) + (m– )γr,m–(x). (.)

Thus,

γ_r_,_m(x) =rγr–,m–(x) + (m– )γr,m–(x) (m≥r), (.)

withγr,r–(x) = .

Replacingmbym+ , we get

mγr,m(x) =γr,m+(x) –rγr–,m(x). (.)

Denoting_γr,m(x)dxbyar,m, we have

ar,m= –

r

mar–,m+ 

m r,m+, (.)

where r,m=γr,m() –γr,m(). From the recurrence relation (.), we can easily show that





γr,m(x)dx= r–

j=

(–)j–(r)j–

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r,m=γr,m() –γr,m()

=

c+c+···+cr=m,c,...,cr≥

Bc()· · ·Bcr()

c· · ·cr

–

c+c+···+cr=m,c,...,cr≥

Bc· · ·Bcr

c· · ·cr

=

c+c+···+cr=m,c,...,cr≥

(Bc+δ,c)· · ·(Bcr+δ,cr)

–

c+c+···+cr=m,c,...,cr≥

Bc· · ·Bcr

c· · ·cr

=

≤a≤r–

r a

c+c+···+ca=m+a–r,c,...,ca≥

cc· · ·ca

. (.)

Observe here that the sum over all positive integersc, . . . ,crsatisfyingc+c+· · ·+cr=

mof any term withaofBceandbofδ,cf (≤e,f≤r,a+b=r), all give the same sum

c+c+···+cr=m,c,...,ca≥

Bc· · ·Bcaδ,ca+· · ·δ,ca+b

cc· · ·cr

=

c+c+···+ca=m+a–r,c,...,ca≥

cc· · ·ca

, (.)

and that, asm+a–r≥a, there are no empty sums.

Here we note that, fora= , the inner sum isδm,rsince it corresponds to the sums

c+c+···+cr=m,c,...,cr≥

δ,cδ,c· · ·δ,cr

cc· · ·cr

. (.)

Also,γr,m() =γr,m()⇔ r,m= .

Now, we would like to consider the function

γr,m

x=

c+c+···+cr=m,c,...,cr≥  cc· · ·cr

Bc

xBc

x· · ·Bcr

x, (.)

deﬁned on (–∞,∞), which is periodic with period . The Fourier series ofγr,m(x) is

∞

n=–∞

C_n(r,m)eπinx_, _(.)

where

C_n(r,m)=



 γr,m

xe–πinxdx=





γr,m(x)e–πinxdx. (.)

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Case :n= .

From this, we obtain

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Also, we note that, forn= ,

C_n(,m)=  m





Bm(x)e–πinxdx= –

(m– )!

(πin)m. (.)

Thus, forn= , (.) together with (.) determine allCn(r,m)recursively.

Case :n= .

C_(r,m)=





γr,m(x)dx= r

j=

(–)j–(r)j–

mj r–j+,m+. (.)

γr,m(x) (m≥r≥) is piecewiseC∞. In addition,γr,m(x) is continuous for those

pos-itive integersr,mwith r,m=  and discontinuous with jump discontinuities at integers

for those positive integersr,mwith r,m= .

Assume ﬁrst that r,m=  for some integersr,mwithm≥r≥. Thenγr,m() =γr,m().

Henceγr,m(x) is piecewiseC∞and continuous. Thus the Fourier series ofγm(x)

con-verges uniformly toγm(x), and

γm

x=C(r,m)+ ∞

n=–∞

n=

C_n(r,m)eπinx,

whereC_(r,m)is given by (.), andCn(r,m), for eachn= , are determined by relations (.)

and (.).

Now, we are ready to state our ﬁrst theorem.

Theorem . For all integers s,l,with l≥s≥,we let

s,l=

≤a≤s–

s a

c+···+ca=l+a–s,c,...,ca≥

Bc· · ·Bca

c· · ·ca

=δs,l+

≤a≤s–

s a

c+c+···+ca=l+a–s,c,...,ca≥

Bc· · ·Bca

c· · ·ca

. (.)

Assume that r,m= for some integers r,m with m≥r≥.Then we have the following.

c+c+···+cr=m,c,...,cr≥



c···crBc(x)· · ·Bcr(x)has the Fourier series expansion

c+c+···+cr=m,c,...,cr≥  c· · ·cr

Bc

x· · ·Bcr

x

=C_(r,m)+ ∞

n=–∞,n=

C(_nr,m)eπinx_,

where C(r,m)=

r–

j=(–)j– (

r)j–

mj r–j+,m+,with C(,m)= ,and C (r,m)

n ,for each n= ,are

de-termined recursively from

C_n(r,m)=

m–r+

j=

r(m– )j– (πin)j C

(r–,m–j)

n –

m–r+

j=

(m– )j–

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and

C_n(,m)= –(m– )!

(πin)m. (.)

Here the convergence is uniform.

Next, assume that r,m=  for some integersr,mwithm≥r≥. Thenγr,m()=γr,m().

Henceγr,m(x) is piecewiseC∞and discontinuous with jump discontinuities at integers.

Then the Fourier series ofγr,m(x) converges pointwise toγr,m(x) forx∈/Zand

con-verges to

 

γr,m() +γr,m()

=γr,m() +

  r,m

=

c+c+···+cr=m,c,...,cr≥

 c· · ·cr

Bc· · ·Bcr+



 r,m (.)

forx∈Z.

Theorem . For all integers s,l with l≥s≥,we let

s,l=

≤a≤s–

s a

c+c+···+ca=l+a–s,c,...,ca≥

Bc· · ·Bca

c· · ·ca

=δs,l+

≤a≤s–

s a

c+c+···+ca=l+a–s,c,...,ca≥

Bc· · ·Bca

c· · ·ca

. (.)

Assume that r,m= for some integers r,m with m≥r≥.Let C(r,m),C (r,m)

n (n= )be

as in Theorem..Then we have the following.

C_(r,m)+ ∞

n=–∞,n=

C_n(r,m)eπinx

=

⎧ ⎨ ⎩

c+c+···+cr=m,c,...,cr≥



c···crBc(x)· · ·Bcr(x) for x∈/Z,

c+c+···+cr=m,c,...,cr≥c···crBc· · ·Bcr+_ r,m for x∈Z.

(.)

Acknowledgements

The third author is appointed as a chair professor at Tianjin Polytechnic University by Tianjin City in China from August 2015 to August 2019.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the manuscript and typed, read, and approved the ﬁnal manuscript.

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Received: 28 February 2017 Accepted: 29 July 2017 References

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