# Bernoulli Identities and Combinatoric Convolution Sums with Odd Divisor Functions

Authors: Daeyeoul Kim; Yoon Kyung Park
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We study the combinatoric convolution sums involving odd divisor functions, their relations to Bernoulli numbers, and some interesting applications.


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## Body

## 1. Introduction

The Bernoulli polynomialsBk(x), which are usually defined by the exponential generating function
(1)textet-1=∑k=0∞‍Bk(x)tkk!,
play an important role in different areas of mathematics, including number theory and the theory of finite differences. It is well known that Bk=Bk(0) are rational numbers. It can be shown that B2k+1=0 for k≥1 and is alternatively positive and negative for even k. The Bk are called Bernoulli numbers. Let ℕ denote the set of positive integers. Further, let N,d,k∈ℕ, where s,r∈ℕ∪{0}. Throughout this paper, we define divisor functions as follows:
(2)σs(n)=∑d∣n‍ds,σs,r(n;m)=∑d∣nd≡rmodmds,σ~s(n)=∑d∣n‍(-1)d-1ds,σs*(n):=∑d∣nn/doddds.
We also make use of the following convention:
(3)σs(n)=0ifn∉ℤorn≤0.Ramanujan [1] proved that
(4)∑m=1n-1σ1(m)σ1(n-m)=112(5σ3(n)+(1-6n)σ1(n)),∑m=1n-1σ1(m)σ3(n-m)=1240[21σ5(n)+(10-30n)σ3(n)-σ1(n)]
using elementary arguments.Letℌ be the complex upper half plane {z∈ℂ:Im(z)>0} and let q be e2πiz for z∈ℌ. Denote η(z) by the Dedekind η-function
(5)η(z)=q1/24∏n≥1‍(1-qn)
and c6(n) by the nth coefficient of (η(z)η(2z)η(3z)η(6z))2. Alaca and Williams [2] proved that
(6)∑(l,m)∈ℕ22l+3m=nσ1(l)σ1(m)=1120σ3(n)+130σ3(n2)+340σ3(n3)+310σ3(n6)+(124-n12)σ1(n2)+(124-n8)σ1(n3)-1120c6(n).
It turns out that we need not only divisor functions but also the coefficients of certain modular functions. For other divisor functions, Hahn [3] showed that
(7)16∑m=1n-1σ~1(m)σ~3(n-m)=-σ~5(n)+2(n-1)σ~3(n)+σ~1(n)
and Glaisher [4–6] extended Besgue’s formula by replacing σ1(n) in the convolution sum in (4) by other sums σ*(n); for example,
(8)24∑m=1n-1σ1*(m)σ1*(n-m)=6σ3(n)-6σ3(n2)-6nσ1(n)+6nσ1(n2).
Recently, the combinatorial convolution sum is studied [7–10]. In [10] Williams proved the following.Proposition 1.
Letk,n∈ℕ and n≥2. Then
(9)∑s=0k-1(2k2s+1)∑m=1n-1σ2k-2s-1(m)σ2s+1(n-m)=2k+34k+2σ2k+1(n)+(k6-n)σ2k-1(n)+12k+1∑j=2k(2k+12j)B2jσ2k+1-2j(n).Cho et al. found out the linear sum for combinatorial convolution sum ofσl(m;i,N) in [7].Proposition 2.
Fork,N∈ℕ and N≥3, one has
(10)∑r=02k(2kr)‍∑i=1[(N-1)/2]‍∑m=1n-1‍σ2k-r(m;i,N)σr(n-m;i,N)=σ2k+1*(n;N)-2Nnσ2k-1*(n;N)-1N∑i=1[(N-1)/2]‍(N-2i)σ2k(n;i,N)-1+(-1)N2(σ2k+1*(2nN;2)-2Nnσ2k-1*(2nN;2)),
where
(11)σr(n;i,N)=∑d|nn/d≡i(N)‍dr-(-1)r∑d|nn/d≡-i(N)‍dr,σr*(n;N)=∑d|nn/d≢0(N)‍dr=σr(n)-σr(nN).DenoteB~k by ∑j=0k‍(kj)2jBj=2kBk(1/2). The generating function 2tet/(e2t-1)  of B~k is an even function and B~k is zero for all odd positive integer k. The aim of this paper is to study two combinatorial convolution sums of the analogous type of Proposition 2. When we write the convolution sums as linear sum of divisor function, in the result by Williams the coefficients are Bk=Bk(0) and ours are Bk(1/2). More precisely, we prove the following theorems.Theorem 3.
Fork,n∈ℕ and n≥2,
(12)∑s=0k-1‍(2k2s+1)∑m=1n-1‍σ2k-2s-1,1(m;2)σ2s+1,1(n-m;2)=22k-1σ2k+1(n2)+14k+2∑i=0k(2k+12i+1)×B~2k-2iσ2i+1,1(n;2).Equation (7) is a special case when k=2 for the following theorem because B~4=14/30 and B4=-1/30.Theorem 4.
Fork,n∈ℕ and n≥2,
(13)∑s=0k-1‍(2k2s+1)∑m=1n-1σ~2k-2s-1(m)σ~2s+1(n-m)‍=-12σ~2k+1(n)+(n-k2)σ~2k-1(n)+14k+2∑i=0k-2‍(2k+12i+1)×(B~2k-2i+(22k-2i-2)B2k-2i)σ2i+1(n)+14k+2∑i=0k-2‍(2k+12i+1)×(B~2k-2i-22k-2iB2k-2i)σ~2i+1(n).Remark 5.
The product of two modular forms is another modular form of bigger weight. The dimension[(k+1)/2]+1 of space M2k+2(Γ0(2)) of modular forms on Γ0(2) is approximately linear for k and the space E2k+2(Γ0(2)) generated by generating functions of divisor functions is clearly 2 as k grows. More precisely speaking, for the Eisenstein series G2k+2 and G2k+2* which will be defined in Section 2(14)G2s+2*(z)G2k-2s*(z)∈M2k+2(Γ0(2))=E2k+2(Γ0(2))⨁S2k+2(Γ0(2)),
where S2k+2(Γ0(2)) is the space of cusp form of weight 2k+2 on Γ0(2) and it is orthogonal complement of  E2k+2(Γ0(2)) in  M2k+2(Γ0(2)). Since  E2k+2(Γ0(2)) = 〈G2k+2,G2k+2*〉 and S2k+2(Γ0(2)) = 〈h1,…,h[(k+1)/2]-1〉,
(15)G2s+2*(z)G2k-2s*(z)=α1G2k+2(z)+α2G2k+2*(z)+∑r=1[(k+1)/2]-1βrhr(z)
for suitable constants  α1,α2,β1,…,β[(k+1)/2]-1. On the other hand, Theorems 3 and 4 show that the combinatorial convolution sums are written as only divisor functions; that is,
(16)∑s=0k-1(2k2s+1)G2s+2*(z)G2k-2s*(z)=α1G2k+2(z)+α2G2k+2*(z).
The disappearance of β1,…,β[(k+1)/2]-1 is observed in Examples 17 and 18.
All calculations in Lemmas6 and 7 and Theorems 9 and 10 are obtained by usins  SAGE.


## 2. Modular Forms

In this section, we observe the convolution sums as a view point of generating functions of divisor functions.The normalized Eisenstein seriesGk is defined by
(17)Gk(z)=-Bk2k+∑n=1∞‍σk-1(n)qn.For the generating function ofσk-1,1(n) we denote
(18)Gk*(z):=Gk(z)-2k-1Gk(2z)=2k-1-12kBk+∑n=1∞‍σk-1,1(n;2)qn.LetΓ be a finite index subgroup of  SL(2,ℤ). The modular formf(z) of weight k on Γ is a holomorphic function on ℌ such that
(19)f(az+bcz+d)=(cz+d)kf(z)∀(abcd)∈Γ
for a positive integer k. The vector space over ℂ of holomorphic modular forms of weight k on Γ is finite dimensional and is denoted by Mk(Γ).Note thatGk∈Mk(SL(2,ℤ)) (if 2∣k>2) and Gk*∈Mk(Γ0(2)) (if 2∣k≥2) for Γ0(2)={(abcd)∈SL(2,ℤ):2∣c}. Moreover, the product of two modular forms f1andf2 of weights k1andk2 is also modular form of weight k1+k2.TheΔ is the discriminant function
(20)Δ(z)=η24(z)=∑n≥1‍τ(n)qn
with Ramanujan τ-function as its coefficient. It is modular of weight 12 on SL(2,ℤ).Define the following two weight14 modular forms Δ14,1 and Δ14,2 by using the Dedekind η-function defined in (5):
(21)Δ14,1(z)=14(G6*(z)-5G6(z))η8(z)η8(2z)=∑N=1∞‍c14,1(N)qN,Δ14,2(z)=243(G2*(z))3η8(z)η8(2z)=∑N=1∞‍c14,2(N)qN.We get the lemma.Lemma 6.
Consider the following:(1)
(22)G2*(z)G10*(z)=315528G12(z)-22760808G12*(z)+1365691Δ(z)+26880691Δ(2z),(2)
(23)G4*(z)G8*(z)=1744224G12(z)-79221120G12*(z)-203691Δ(z)-4928691Δ(2z),(3)
(24)(G6*(z))2=15528G12(z)-61348264G12*(z)+85691Δ(z)+2560691Δ(2z).Proof.
The functionsG2*G10*, G4*G8*, and (G6*)2 are modular functions of weight 12 on Γ0(2). Note that M12(Γ0(2)) is a 4-dimensional vector space over ℂ generated by G12(z),G12*(z),Δ(z), and Δ(2z) because dimℂ(Mk(Γ0(2)))=[k/4]+1. Their q-expansions are as follows:
(25)G12(z)=69165520+q+2049q2+177148q3+4196353q4+48828126q5+O(q6),G12*(z)=-141447765520+q+q2+177148q3+q4+48828126q5+O(q6),Δ(z)=q-24q2+252q3-1472q4+4830q5+O(q6),G2*(z)G10*(z)=5116336+8744q+13144q2+912911q3+86640344q4+222380122q5+O(q6),G4*(z)G8*(z)=889115200-47160q+113160q2-276940q3+354673160q4-752180q5+O(q6),(G6*(z))2=961254016+31252q+283252q2+201763q3+123259252q4+3673142q5+O(q6).
By comparing the above expansions with ones of G2s*(z)G12-2s*(z) for s=1,2,3, we get our result.Lemma 7.
Consider the following:(1)
(26)G2*(z)G12*(z)=691174752G14(z)-1885739715609440G14*(z)-89186Δ14,1(z)-8525762Δ14,2(z),(2)
(27)G4*(z)G10*(z)=31174752G14(z)-476928834080G14*(z)+8186Δ14,1(z)+245254Δ14,2(z),(3)
(28)G6*(z)G8*(z)=17349504G14(z)-5249110093760G14*(z)-986Δ14,1(z)-25254Δ14,2(z).Proof.
One can prove these by using a similar way to Lemma6 and 4-dimensional vector space M14(Γ0(2)) generated by G14,G14*,Δ14,1, and Δ14,2 over ℂ because their q-expansions are as follows:
(29)G14(z)=-124+q+8193q2+1594324q3+67117057q4+1220703126q5+O(q6),G14*(z)=819124+q+q2+1594324q3+q4+1220703126q5+O(q6),Δ14,1(z)=q-64q2-1836q3+4096q4+3990q5+O(q6),Δ14,2(z)=q+64q2+1236q3+4096q4-57450q5+O(q6).Remark 8.
(1)Δ(z) is the normalized Hecke eigenform on the full modular group of weight 12:
(30)τ(pr+1)=τ(pr)τ(p)-p11τ(pr-1)forpprime,r≥1,τ(mn)=τ(m)τ(n)ifgcd(m,n)=1.
(2)Δ14,1(z) and Δ14,2(z) are normalized newforms on Γ0(2) of weight 14. The coefficients ck,j(n) (j=1,2) satisfy that
(31)c14,j(pr+1)=c14,j(pr)c14,j(p)-p13c14,j(pr-1)forpoddprimep,r≥1,c14,j(2r)=c14,j(2)rforr≥0,c14,j(mn)=c14,j(m)c14,j(n)ifgcd(m,n)=1.
Moreover, for the Atkin-Lehner involution W2=(0-121),
(32)Δ14,1∣14W2=Δ14,1,Δ14,2∣14W2=-Δ14,2
if we define the action γ=(abcd)∈GL(2,ℚ) on the complex valued function f as
(33)(f∣kγ)(z)=det(γ)k/2(cz+d)-kf(az+bcz+d).
By the help of Lemmas6 and 7 we get the formulae for each convolution sum.Theorem 9.
Consider the following:(1)
(34)∑m=1n-1‍σ1,1(m;2)σ9,1(n-m;2)=315528σ11(n)-22760808σ11,1(n;2)-124σ9,1(n;2)-511264σ1,1(n;2)+1365691τ(n)+26880691τ(n2),(2)
(35)∑m=1n-1‍σ3,1(m;2)σ7,1(n-m;2)=1744224σ11(n)-79221120σ11,1(n;2)+7240σ7,1(n;2)+127480σ3,1(n;2)-203691τ(n)-4928691τ(n2),(3)
(36)∑m=1n-1‍σ5,1(m;2)σ5,1(n-m;2)=15528σ11(n)-61348264σ11,1(n;2)-31252σ5,1(n;2)+85691τ(n)+2560691τ(n2).Theorem 10.
Consider the following:(1)
(37)∑m=1n-1‍σ1,1(m;2)σ11,1(n-m;2)=691174752σ13(n)-1885739715609440σ13,1(n;2)-124σ11,1(n;2)+141447765520σ1,1(n;2)-89186c14,1(n)-8625762c14,2(n),(2)
(38)∑m=1n-1‍σ3,1(m;2)σ9,1(n-m;2)=31174752σ13(n)-476928834080σ13,1(n;2)+7240σ9,1×(n;2)-511264σ3,1(n;2)-8186c14,1(n)+245254c14,2(n),(3)
(39)∑m=1n-1‍σ5,1(m;2)σ7,1(n-m;2)=17349504σ13(n)-5249110093760σ13,1(n;2)-31504σ7,1(n;2)+127480σ5,1(n;2)-986c14,1(n)-25254c14,2(n).


## 3. Proof of Theorems

In his series of eighteen papers published between 1858 and 1865, Joseph Liouville (1809–1882) stated without proof several elementary arithmetic formulae. One of these is the following formula.Proposition 11 (see [10, page 112]).
Letf:ℤ→ℂ be an even function. Let n∈ℕ. Then one obtains
(40)∑(a,b,x,y)∈ℕ4ax+by=na,bodd‍(f(a-b)-f(a+b))=f(0)(σ1*(n)-σ0*(n))-12∑d∈ℕd∣ndeven‍df(d)-2∑d∈ℕd∣ndodd‍∑t∈ℕ1≤t<dteven‍f(t).Now we are ready to prove our theorems in Section1.Proof of Theorem3.
We applyf(x)=x2k (k≥1) in Proposition 11. Then the left-hand side is
(41)∑(a,b,x,y)∈ℕ4ax+by=na,bodd(a-b)2k-(a+b)2k=∑(a,b,x,y)∈ℕ4ax+by=na,bodd(∑r=02k‍(2kr)(-1)ra2k-rbrhhhhhhhhhhhhh-∑r=02k‍(2kr)a2k-rbr)=-2∑(a,b,x,y)∈ℕ4ax+by=na,bodd∑r=0rodd2k‍(2kr)a2k-rbr=-2∑s=0k-1‍(2k2s+1)∑(a,b,x,y)∈ℕ4ax+by=na,bodda2k-2s-1b2s+1=-2∑s=0k-1(2k2s+1)‍hhhhh×∑m=1n-1(∑a∣m2∤a‍a2k-2s-1∑b∣(n-m)2∤b‍b2s+1)(∑b∣(n-m)2∤b‍b2s+1)‍=-2∑m=1n-1‍∑s=0k-1‍(2k2s+1)hhhhhhhhhhh×σ2k-2s-1,1(m;2)σ2s+1,1(n-m;2).
On the other hand by using Bernoulli’s identity [10, page 42],
(42)∑l=1n-1‍lk=1k+1∑j=0k‍(k+1j)Bjnk+1-j,
the right-hand side is
(43)∑(a,b,x,y)∈ℕ4ax+by=na,bodd(a-b)2k-(a+b)2k=∑(a,b,x,y)∈ℕ4ax+by=na,bodd(∑r=02k‍(2kr)(-1)ra2k-rbrhhhhhhhhhhhhh-∑r=02k‍(2kr)a2k-rbr)=-2∑(a,b,x,y)∈ℕ4ax+by=na,bodd∑r=0rodd2k‍(2kr)a2k-rbr=-2∑s=0k-1‍(2k2s+1)∑(a,b,x,y)∈ℕ4ax+by=na,bodda2k-2s-1b2s+1=-2∑s=0k-1(2k2s+1)‍hhhhh×∑m=1n-1(∑a∣m2∤a‍a2k-2s-1∑b∣(n-m)2∤b‍b2s+1)(∑b∣(n-m)2∤b‍b2s+1)‍=-2∑m=1n-1‍∑s=0k-1‍(2k2s+1)hhhhhhhhhhh×σ2k-2s-1,1(m;2)σ2s+1,1(n-m;2).
After dividing both sides by -2 we get
(44)22k-1σ2k+1(n2)+14k+2∑i=0k‍B~2k-2i(2k+12i+1)σ2i+1,1(n;2).Remark 12.
Whenf is a constant function in Proposition 11, it is a trivial formula such that
(45)σ1(n)=2σ1*(n)-σ1,1(n;2).Corollary 13.
Fork≥1,
(46)∑r=02k+1‍∑j=02k+1-r(2k+1j,r,2k+1-r-j)‍2jBj=0.Proof.
Applyingn=1 in Theorem 3, our result is proved.Corollary 14.
LetN∈ℕ. Then one has
(47)∑s=0k-1‍(2k2s+1)∑m=12∤m2N‍σ2k-2s-1(m)σ2s+1(2N-m)=22k-1σ2k+1*(N).Proof.
Letn=2N in Theorem 3. Then the left-hand side is
(48)∑s=0k-1‍(2k2s+1)∑m=12N-1‍σ2k-2s-1,1(m;2)σ2s+1,1(2N-m;2)=∑s=0k-1‍(2k2s+1)mmim×[∑m=12∣m2N-1‍σ2k-2s-1,1(m;2)σ2s+1,1(2N-m;2)mmnmm+∑m=12∤m2N-1‍σ2k-2s-1,1(m;2)σ2s+1,1(2N-m;2)∑m=12∣m2N-1‍]=∑s=0k-1‍(2k2s+1)mmm×[∑m=12∤m2N-1‍∑m=1N-1‍σ2k-2s-1,1(2m;2)σ2s+1;2(2N-2m)mmmmm+∑m=12∤m2N-1‍σ2k-2s-1,1(m;2)σ2s+1,1(2N-m;2)∑m=1N-1‍]=∑s=0k-1‍(2k2s+1)∑m=1N-1‍σ2k-2s-1,1(m;2)σ2s+1,1(N-m;2)mm+∑s=0k-1‍(2k2s+1)∑m=12∤m2N‍σ2k-2s-1(m)σ2s+1(2N-m).
The right-hand side of Theorem 3 for n=2N is
(49)22k-1σ2k+1(N)+14k+2∑i=0k‍(2k+12i+1)B~2k-2iσ2i+1,1(2N;1)=22k-1σ2k+1(N)+14k+2×∑i=0k(2k+12i+1)‍B~2k-2iσ2i+1,1(N;2)=22k-1σ2k+1(N)-22k-1σ2k+1(N2)+∑i=0k-1‍(2k2i+1)∑m=1N-1‍σ2k-2i-1,1(m;2)σ2i+1,1(N-m;2).
Sinceσs*(N)=σs(N)-σs(N/2), we are done.Remark 15.
The above result is also in [8, Theorem 3.4], but we do not use the combinatoric convolution sums of σk*-functions but odd divisor function.Corollary 16.
For the odd primep case, one has
(50)∑s=0k-1‍(2k2s+1)∑m=1p-1‍σ2k-2s-1,1(m;2)σ2s+1,1(p-m;2)=14k+2∑r=02k+1‍∑j=02k+1-r‍(2k+1j,r,2k+1-j-r)2jprBj.Proof.
In Theorem3,
(51)∑s=0k-1‍(2k2s+1)∑m=1p-1‍σ2k-2s-1,1(m;2)σ2s+1,1(p-m;2)=14k+2∑s=0k‍B~2k-2s(2k+12s+1)σ2s+1,1(p;2)=14k+2∑r=02k+1‍B~2k-r+1(2k+1r)σr(p)=14k+2∑r=02k+1‍∑j=02k+1-r‍(2k+1j,r,2k+1-j-r)2jBj(1+pr)=14k+2∑r=02k+1‍∑j=02k+1-r‍(2k+1j,r,2k+1-j-r)2jprBj
by Corollary 13.Proof of Theorem4.
Note that(52)σ~s(n)=σs(n)-2s+1σs(n2),σs,1(n;2)=σs(n)-2sσs(n2).
We reconsider Proposition 1 as the last one in the previous line:
(53)∑s=0k-1‍(2k2s+1)∑m=1n-1‍σ2k-2s-1(m2)σ2s+1(n-m2)=2k+34k+2σ2k+1(n2)+(k6-n2)σ2k-1(n2)+12k+1∑j=2k‍(2k+12j)B2jσ2k+1-2j(n2).
Thus,(54)∑s=0k-1‍(2k2s+1)∑m=1n-1‍σ~2k-2s-1(m)σ~2s+1(n-m)=∑s=0k-1‍(2k2s+1)∑m=1n-1‍(σ2k-2s-1(m)-22k-2sσ2k-2s-1(m2))×(σ2s+1(n-m)-22s+2σ2s+1(n-m2))=2∑s=0k-1‍(2k2s+1)∑m=1n-1‍σ2k-2s-1,1(m;2)σ2s+1,1(n-m;2)-∑s=0k-1‍(2k2s+1)∑m=1n-1‍σ2k-2s-1(m)σ2s+1(n-m)+22k+1∑s=0k-1‍(2k2s+1)∑m=1n-1‍σ2k-2s-1(m2)σ2s+1(n-m2)=-12σ~2k+1(n)+(n-k2)σ~2k-1(n)+14k+2×∑i=0k-2‍(2k+12i+1)(B~2k-2i+(22k-2i-2)B2k-2i)σ2s+1(n)+14k+2∑i=0k-2‍(2k+12i+1)(B~2k-2i-22k-2iB2k-2i)σ~2i+1(n)
by Proposition 1, Theorem 3, and (53).The following examples show us that the coefficients of cusp forms disappear in the combinatorial convolution sum for the weights12 and 14, explicitly.Example 17.
Consider the case of Theorem3 for k=5. The left-hand side is
(55)∑s=04‍(102s+1)∑m=1n-1‍σ9-2s,1(m;2)σ2s+1(n-m;2)=20∑m=1n-1‍σ1,1(m;2)σ1,1(n-m;2)+240∑m=1n-1‍σ3,1(m;2)σ7,1(n-m;2)+252∑m=1n-1σ5,1(m;2)σ5,1(n-m;2).‍
When we put the formula in Theorem9 in the above, the coefficients of τ(n) and τ(n/2) are zero.Example 18.
The left-hand side of Theorem3 when k=6 is
(56)24∑m=1n-1‍σ1,1(m;2)σ11,1(n-m;2)+440∑m=1n-1‍σ3,1(m;2)σ9,1(n-m;2)+1584∑m=1n-1‍σ5,1(m;2)σ7,1(n-m;2).
By applying Theorem 10 to the above, one can check that our theorem is true and the coefficients c14,1(n) and c14,2(n) of cusp forms disappear.


## Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.



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