Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 181264, 14 pages
doi:10.1155/2012/181264
Research Article
Eisenstein Series Identities Involving the Borweins’
Cubic Theta Functions
Ernest X. W. Xia and Olivia X. M. Yao
Department of Mathematics, Jiangsu University, Jiangsu, Zhenjiang 212013, China
Correspondence should be addressed to Olivia X. M. Yao, yaoxiangmei@163.com
Received 28 March 2012; Revised 15 May 2012; Accepted 23 May 2012
Academic Editor: Ferenc Hartung
Copyright q 2012 E. X. W. Xia and O. X. M. Yao. 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.
Based on the theories of Ramanujan’s elliptic functions and the p, k-parametrization of theta
functions due to Alaca et al. 2006, 2007, 2006 we derive certain Eisenstein series identities
involving the Borweins’ cubic theta functions with the help of the computer. Some of these
identities were proved by Liu based on the fundamental theory of elliptic functions and some
of them may be new. One side of each identity involves Eisenstein series, the other products of
the Borweins’ cubic theta functions. As applications, we evaluate some convolution sums. These
evaluations are different from the formulas given by Alaca et al.
1. Introduction
Let N and C denote the sets of positive integers and complex numbers, respectively.
Throughout the paper, we always assume that q ∈ C and |q| < 1.
In their paper, J. M. Borwein and P. B. Borwein 1 introduced the following three
functions:
a q ∞
qm
2
mnn2
,
m,n−∞
b q ∞
ωm−n qm mnn ,
2
2
1.1
m,n−∞
c q ∞
2
2
qm1/3 m1/3n1/3n1/3 ,
m,n−∞
2
Journal of Applied Mathematics
where ω e2πi/3 . These functions are now called the Borweins’ cubic theta functions. The
Borwein brothers 1 derived representations for bq and cq in terms of infinite products,
namely,
3
q; q
b q 3 3 ∞ ,
q ;q ∞
3
3q1/3 q3 ; q3 ∞
,
c q q; q ∞
1.2
1.3
where
∞ 1 − aqi .
a; q ∞ 1.4
i0
The function aq has the following representation derived by the Borwein brothers 1 and
Berndt 2:
4 4 12 12 2 2 q ;q
q ;q
2
2 2
3 6
6 6
a q −q; q
q ;q
−q ; q
q ;q
4q 2 4 ∞ 6 12 ∞ .
∞
∞
∞
∞
q ;q ∞ q ;q ∞
1.5
Elementary proofs of 1.2, 1.3, and 1.5 can be found in 3. The Borwein brothers 1 also
proved the following well-known relation for aq, bq and cq, namely:
a3 q b3 q c3 q .
1.6
In his second notebook 4, Ramanujan gave the definitions of the Eisenstein series
Lq, Mq, and Nq, namely,
∞
nqn
,
L q : 1 − 24
1 − qn
n1
1.7
∞
n3 qn
M q : 1 240
,
1 − qn
n1
1.8
∞
n5 qn
N q : 1 − 504
.
1 − qn
n1
1.9
Journal of Applied Mathematics
3
Utilizing Ramanujan’s elliptic functions in the theory of signature 3, Berndt et al. 5
proved the following representations for Mq, Mq3 , Nq, and Nq3 , namely:
M q a q a3 q 8c3 q ,
1 M q3 a q 9a3 q − 8c3 q ,
9
6
N q a q − 20a3 q c3 q − 8c3 q ,
4 8 N q3 a6 q − a3 q c3 q c3 q .
3
27
1.10
1.11
1.12
1.13
Chan 6 proved 1.10 and 1.11 by employing the classical theory of elliptic functions and
modular equations of degree 3. Based on the fundamental theory of elliptic functions, Liu
7, 8 provided different proofs of 1.10, 1.11, 1.12, and 1.13. He also discovered some
striking Eisenstein series identities.
In this paper, using the parameters p and k introduced by Alaca et al. 9–11 see 2.1,
we deduce some Eisenstein series identities involving the Borweins’ cubic theta functions
with the help of the computer. These identities are examples of sum-to-product identities.
Some of these identities were proved by Liu 7, 8 based on the fundamental theory of elliptic
function and some of them may be new.
This paper is organized as follows. In Section 2, we gave the p, k-parametrization
of mLqm − Lq, and Mql for m ∈ {2, 3, 4, 6, 12}, l ∈ {1, 2, 3, 6}, which are due to Alaca
et al. 9–11. Section 3 is devoted to deriving some Eisenstein series identities involving the
Borweins’ cubic theta functions. As corollaries of our results, in Section 4, we derive new
representations for the convolution sums i6jn σiσj and i12jn σiσj and contrast
them with the known evaluations due to Alaca and Williams 11 and Alaca et al. 10.
2. Eisenstein Series Lq, Mq and Parameters p, k
In this section, we gave the parametric representations for mLqm − Lq, Mql , bqi and
cqi for m ∈ {2, 3, 4, 6, 12}, l ∈ {1, 2, 3, 6} and i ∈ {1, 2, 4} in terms of the parameters p and k
first defined by Alaca and Williams 11, namely,
ϕ2 q − ϕ2 q3
,
pp q 2ϕ2 q3
2.1
ϕ3 q3
kk q .
ϕ q
Alaca and Williams 11 derived the representations of Mq, Mqi , and iLqi −
Lq i 2, 3, 6 in terms of p and k. Equations 3.69, 3.70, 3.71, 3.72, 3.84, 3.87,
and 3.89 in 11 are
M q 1 124p 964p2 2788p3 3910p4 2788p5 964p6 124p7 p8 k4 ,
M q2 1 4p 64p2 178p3 235p4 178p5 64p6 4p7 p8 k4 ,
4
Journal of Applied Mathematics
M q3 1 4p 4p2 28p3 70p4 28p5 4p6 4p7 p8 k4 ,
M q6 1 4p 4p2 − 2p3 − 5p4 − 2p5 4p6 4p7 p8 k4 ,
2L q2 − L q 1 14p 24p2 14p3 p4 k2 ,
3L q3 − L q 2 16p 36p2 16p3 2p4 k2 ,
6L q6 − L q 5 22p 36p2 22p3 5p4 k2 ,
2.2
respectively. Alaca et al. 12, 13 also deduced the representations of 12Lq12 − Lq, Mq12 and 4Lq4 − Lq in terms of p and k. Equations 3.12, and 3.19 in 12 and 3.13 in 13
are
12L q12 − L q 11 34p 36p2 16p3 2p4 k2 ,
M q
12
1 6 1 7 1 8 4
2
3
4
5
1 4p 4p − 2p − 5p − 2p p p p k ,
4
4
16
4L q4 − L q 24p3 36p2 18p 3 k2 ,
2.3
respectively. Alaca et al. 9 also derived the following parametric representations for bq
and cq in terms of p and k. From Theorems 1, 2, and 4 in 9, we have
1/3
1 − p 1 − p 1 2p 2 p
k
,
b q 1/3
2
1/3
k
3 1p p 1p
,
c q 1/3
2
1 − p1 2p2 p2/3 k
,
b q2 22/3
3p1 p2/3 k
,
c q2 22/3
2 p1 − p1 2p2 p1/3 k
,
b q4 24/3
3pp1 p1/3 k
c q4 .
24/3
2.4
We now describe our approach. Let Req, eq2 , eq4 be a function, where e b or e c.
Utilizing the representations for bq, bq2 , bq4 , cq, cq2 and cq4 in terms of p and k,
we derive the representations Rp, k for Req, eq2 , eq4 in terms of p and k. We select
Journal of Applied Mathematics
5
suitable Req, eq2 , eq4 such that Rp, k is a polynomial in p and k. We want to show
that
s
R p, k Ci Ti ,
2.5
i1
where each Ci is a rational number and Ti is a product involving mLqm − Lq and Mqm .
Substituting the representation for Ti in terms of p and k into 2.5, then both sides of 2.5
are functions in p and k. Equating the coefficients of pi kj on both sides of 2.5, we obtain
some linear equations in Ci . If these equations have a solution, then we can use computer to
solve the equations and determine the values of the Ci . We then obtain some Eisenstein series
identities involving the Borweins’ cubic theta functions.
3. Some Eisenstein Series Identities
In this section, we derive some Eisenstein series identities. In fact, utilizing our method, we
can obtain many identities, here we just list some of them. Our main theorem can be stated
as follows.
Theorem 3.1. One has
∞
1−6
n1
nqn
4nq2n
9nq3n
16nq4n
−
−
n
1−q
1 − q2n 1 − q3n 1 − q4n
3.1
6 3 q2 ; q2 ∞ q12 ; q12 ∞
b2 q b q2
,
3 2 b q4
q4 ; q4 ∞ q3 ; q3 ∞ q6 ; q6 ∞
∞
nqn
nq2n
nq4n
9nq12n
13
−
−
1 − qn 1 − q2n 1 − q4n 1 − q12n
n1
q; q
∞
6 3 q4 ; q4 ∞ q2 ; q2 ∞ q3 ; q3 ∞ b2 q4 b q2
,
2 3 b q
q12 ; q12 ∞ q6 ; q6 ∞ q; q ∞
3 2 2 3
∞
q; q ∞ q ; q ∞
2
nqn
2nq2n
9nq3n
18nq6n
1−3
−
−
b
q b q ,
1 − qn 1 − q2n 1 − q3n 1 − q6n
q3 ; q3 ∞ q6 ; q6 ∞
n0
∞
1 − 12
n1
∞
13
n1
∞
13
n1
nqn
8nq2n
9nq3n
−
1 − qn 1 − q2n 1 − q3n
b4 q
2 2 ,
b q
3.3
3.4
12 q2 ; q2 ∞ q3 ; q3 ∞ q12 ; q12 ∞
b4 q2
3 4 ,
3 4
b q b q
q; q ∞ q4 ; q4 ∞ q6 ; q6 ∞
12 2
q6 ; q6 ∞
∞
2 2 6 3 3 4
q ;q ∞ q ;q ∞
q; q
12 2
q2 ; q2 ∞ q3 ; q3 ∞ b4 q2
6 4 2 ,
b q
q; q ∞ q6 ; q6 ∞
nq4n
nq6n
nq12n
4
36
− 36
1 − q6n
1 − q4n
1 − q12n
2nqn
nq2n
9nq6n
−
−
1 − qn 1 − q2n 1 − q6n
nqn
nq2n
nq3n
−
4
−
9
1 − qn
1 − q2n
1 − q3n
3.2
3.5
3.6
6
Journal of Applied Mathematics
∞
5nq2n
nqn
3nq4n
nq12n
13
−
−
−9
1 − q2n 1 − qn 1 − q4n
1 − q12n
n0
3.7
3 4 4 9 6 6 2
q; q ∞ q ; q ∞ q ; q ∞
b q b3 q4
,
12 12 3 2 2 6 b2 q2
q3 ; q3
q ;q
q ;q
13
∞
∞
∞
2n
n
∞
nq
nq
nq3n
nq4n
20
−
3
−
9
−
16
1 − qn
1 − q2n
1 − q3n
1 − q4n
n1
3 2
q4 ; q4 ∞ q6 ; q6 ∞
b3 q b q4
∞
,
3 6 b2 q2
q3 ; q3 ∞ q12 ; q12 ∞ q2 ; q2 ∞
∞
11n3 qn 16n3 q2n 540n3 q6n
107 3 −
32
2 n1 1 − qn
1 − q2n
1 − q6n
3
−
32
−
∞
5 24
n1
∞
11 3 64 4 n1
q; q
9 nqn
6nq6n
−
1 − qn 1 − q6n
2
n3 qn
2n3 q2n 54n3 q6n
−
−
n
1−q
1 − q2n
1 − q6n
6
q2 ; q2 ∞
2 2
∞
2
b
q b q ,
2
2
q3 ; q3 ∞ q6 ; q6 ∞
q; q
6 3.8
3.9
24 4
q2 ; q2 ∞ q3 ; q3 ∞ b8 q2
12 8 4 ,
b q
q; q ∞ q6 ; q6 ∞
243n3 q3n 1080n3 q6n
−
1 − q3n
1 − q6n
3.10
2
24 6 6 4
∞
q; q ∞ q ; q ∞
b8 q
nqn
6nq6n
−
3 5 24
12 8 4 2 ,
1 − qn 1 − q6n
q2 ; q2 ∞ q3 ; q3 ∞ b q
n1
∞
nqn
nq3n
4nq6n
16nq12n
12
−
−
1 − qn 1 − q3n 1 − q6n 1 − q12n
n1
3.11
3
64
5 24
∞
n1
∞
− 74 − 24
n1
nqn
6nq6n
−
n
1−q
1 − q6n
31n3 qn 128n3 q2n
−
1 − qn
1 − q2n
2
6 6 6 3 4 4 q3 ; q3
q ; q ∞ q ; q ∞ c2 q c q2
2 ∞
,
3
9c q4
q; q ∞ q2 ; q2 ∞ q12 ; q12 ∞
∞
nq3n
nq4n
nq6n
nq12n
−
−
1 − q3n 1 − q4n 1 − q6n 1 − q12n
n1
∞
nqn
2nq2n
−
1 − qn 1 − q2n
n1
6 3 q3 q12 ; q12 ∞ q6 ; q6 ∞ q; q ∞ c2 q4 c q2
,
2 3 9c q
q4 ; q4 ∞ q2 ; q2 ∞ q3 ; q3 ∞
3 3
q q3 ; q3 ∞ q6 ; q6 ∞ c q c q2
nq3n
2nq6n
2 2
,
−
9
1 − q3n 1 − q6n
q; q ∞ q ; q ∞
3.12
3.13
3.14
Journal of Applied Mathematics
∞
n1
∞
14
nq2n
2nq3n
nq6n
−
1 − q2n 1 − q3n 1 − q6n
nqn
nq3n
8nq6n
−
1 − qn 1 − q3n 1 − q6n
n1
∞
7
12 2
q2 q6 ; q6 ∞ q; q ∞
c4 q2
4 6 2 ,
9c q
q2 ; q2 ∞ q3 ; q3 ∞
12 2
q3 ; q3 ∞ q2 ; q2 ∞
c4 q
4 6 2 2 ,
9c q
q; q ∞ q6 ; q6 ∞
nqn
nq2n
nq3n
nq4n
nq6n
nq12n
−
4
−
4
4
−
4
1 − qn
1 − q2n 1 − q3n
1 − q6n
1 − q4n
1 − q12n
n1
3.15
3.16
3.17
12 q6 ; q6 ∞ q; q ∞ q4 ; q4 ∞
c4 q2
q
,
3 3
q2 ; q2 ∞ q3 ; q3 ∞ q12 ; q12 ∞ 9c q c q4
∞
nq3n
nq4n
nq6n
nq12n
−5
3
1 − q3n 1 − q4n
1 − q6n
1 − q12n
n1
2 3 9
q2 ; q2 ∞ q3 ; q3 ∞ q12 ; q12 ∞ c q c3 q4
,
4 4 3 6 6 6
9c2 q2
q; q ∞ q ; q ∞ q ; q ∞
q3
∞
n1
nq3n
nqn
nq6n
nq12n
3
− 20
16
n
3n
6n
1−q
1−q
1−q
1 − q12n
q
2 9 3
q ; q2 ∞ q3 ; q3 ∞ q12 ; q12 ∞
3 4 4 6 6 6
q; q ∞ q ; q ∞ q ; q ∞
2
c q c q4
,
9c2 q2
q
25
864
∞
25
1
−
−
1728 36 n1
1
1728
∞
2
8
−
27 9 n1
1
27
6 6
q ; q3 ∞ q6 ; q6 ∞
2 2 2 2
q; q ∞ q ; q ∞
3
∞
5 24
n1
c q c2 q2
,
81
nqn
6nq6n
−
1 − qn 1 − q6n
n3 qn
3n3 q3n 24n3 q6n
−
1 − qn 1 − q3n
1 − q6n
2
24 4
q4 q6 ; q6 ∞ q; q ∞
c8 q2
,
8 12 81c4 q
q2 ; q2 ∞ q3 ; q3 ∞
2 3 3 24 2 2 4
∞
q ;q ∞ q ;q ∞
c8 q
nqn
6nq6n
5 24
−
8 ,
12 1 − qn 1 − q6n
81c4 q2
q; q ∞ q6 ; q6 ∞
n1
3.20
2
5n3 qn 18n3 q2n 48n3 q3n 186n3 q6n
−
−
1 − qn
1 − q2n
1 − q3n
1 − q6n
3.19
3
2
∞
∞
5n3 qn 12n3 q3n 132n3 q6n
nqn
6nq6n
1
1
−
5 24
−
−
18 n1 1 − qn
864
1 − qn 1 − q6n
1 − q3n
1 − q6n
n1
2
3.18
3.21
3.22
8
Journal of Applied Mathematics
1
108
∞
11 24
n1
∞
2
9 n1
nqn
12nq12n
−
1 − qn 1 − q12n
2
1
−
216
∞
5 24
n3 qn
6n3 q3n 96n3 q6n 96n3 q12n
−
−
1 − qn 1 − q3n
1 − q6n
1 − q12n
n1
−
nqn
6nq6n
−
1 − qn 1 − q6n
2
1
216
3.23
12 6 2
q3 ; q3 ∞ q6 ; q6 ∞ q4 ; q4 ∞ c4 q c2 q2
4 .
2 6
81c2 q4
q; q ∞ q2 ; q2 ∞ q12 ; q12 ∞
Remark 3.2. The identities 1.16 and 1.17 in 8 contain typos, they should be 3.1 and
3.5, respectively.
Proof. We first prove the formula 3.1 by our method. We assume that
C1 2L q2 − L q C2 3L q3 − L q C3 4L q4 − L q
b2 q b q2
6
12
C4 6L q − L q C5 12L q
−L q .
b q4
3.24
Equating the coefficients of k2 , pk2 , p2 k2 , p3 k2 , and p4 k2 on both sides of 3.24, we obtain the
following five equations:
C1 2C2 3C3 5C4 11C5 1,
14C1 16C2 18C3 22C4 34C5 −1,
24C1 36C2 36C3 36C4 36C5 −3,
3.25
14C1 16C2 24C3 22C4 16C5 5,
C1 2C2 5C4 2C5 −2.
Solving the above five equations, we obtain
1
C1 − ,
2
3
C2 − ,
4
C3 1,
C4 0,
C5 0.
3.26
Substituting the above values into 3.24, from 1.2 and 1.7, we obtain 3.1.
Similarly, utilizing the same method, we also derive the following formulas:
b2 q4 b q2
1 4 3 1 2
2L q − L q −
4L q − L q 12L q12 − L q ,
16
32
32
b q
3 3 3 6 1 b q b q2 − 2L q2 − L q −
3L q − L q 6L q − L q ,
8
8
8
4
b q
3 3 3L q − L q ,
2 −2 2L q2 − L q 2
2
b q
Journal of Applied Mathematics
b4 q2
3 6 1 2
−
L
q
−
L
q ,
2L
q
6L
q
16
16
b2 q
b4 q2
3 1 1 2
2L q − L q 3L q3 − L q − 4L q4 − L q
4
8
8
b q b q4
3 3 − 6L q6 − L q 12L q12 − L q ,
4
8
b q b3 q4
3 4 3 5 2
12
2L
q
4L
q
12L
q
−
L
q
−
L
q
−
L
q ,
−
16
32
32
b2 q2
b3 q b q4
3 3 1 4 5 2
2L
q
3L
q
4L
q
−
L
q
−
L
q
−
L
q ,
−
4
8
2
b2 q2
27 3 2
1
11
M q − M q2 M q6 −
6L q6 − L q ,
b2 q b2 q2 160
10
8
32
2
8
b q
2
1
27
3 6
1
2
6
M
q
−
M
q
M
q
6L
q
−
−
L
q ,
320
160
160
64
b4 q
b8 q
2
31 64 2 243 3 6
6
M
q
M
q
M
q
−
108M
q
3
6L
q
−
L
q ,
−
10
5
10
b4 q2
c2 q c q2
1 6 1 1 3
,
3L q − L q −
6L q − L q 12L q12 − L q
4 36
18
9
9c q
c2 q4 c q2
1 4 1 3
−
3L q − L q 4L q − L q
72
96
9c q
1 1 6
6L q − L q −
12L q12 − L q ,
144
288
2
c q c q
1 6 1
1
2L q2 − L q 3L q3 − L q −
6L q − L q ,
9
24
72
72
2
4
c q
1
1 6
1
2
3
−
2L
q
3L
q
6L
q
−
L
q
−
L
q
−
L
q ,
−
48
36
144
9c2 q
c4 q
2 6 1 3
3L
q
6L
q
−
L
q
−
L
q ,
−
18
9
9c2 q2
c4 q2
1 3 3 2
2L q − L q 3L q3 − L q − 4L q4 − L q
4 4
8
8
9c q c q
1 12 1 6
− 6L q − L q 12L q − L q ,
4
8
3 4
c q c q
3 4 1 4L q − L q
2 − 3L q3 − L q −
2
8
32
9c q
3 5 6
6L q − L q −
12L q12 − L q ,
16
32
c3 q c q4
5 6 1 3 3
12
−
3L
q
6L
q
12L
q
−
L
q
−
L
q
−
L
q ,
−
8
4
2
9c2 q2
9
10
Journal of Applied Mathematics
99 3 c2 q c2 q2
9
2
3
M q − M q3 M q6 −
6L q6 − L q ,
81
32
40
40
32
3
279 3 c8 q2
9
2
27
2
3
6
6
M
q
M
q
M
q
M
q
6L
q
−
−
−
L
q ,
160
64
20
160
64
81c4 q
36 c8 q
2
9
3
3
6
6
3
6L
q
−
L
q ,
M
q
−
M
q
M
q
−
10
10
5
81c4 q2
c4 q c2 q2
9
36 36 3
M q − M q3 M q6 − M q12
40
20
5
5
81c2 q4
2 3 2
12 3 6
− 6L q − L q
12L q − L q .
8
4
3.27
From the above identities and 1.2, 1.3, 1.7 and 1.8, we may derive other formulas.
4. Convolution Sums
i6jn
σiσj and
i12jn
σiσj
Let n, k ∈ N. The divisor function σk n is defined by
σk n dk ,
d|n
4.1
where d runs through the positive divisors of n. If n is not a positive integer, set σi n 0. As
usual, we write σn for σ1 n. For all n, the convolution ikjn σiσj has been evaluated
explicitly for k 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 16, 18, and 24, see 10–19. In this section, we also
derive the representations for the convolution sums i6jn σiσj and i12jn σiσj
from the identities in Theorem 3.1. Our representations are different from those derived in
11, 12. In fact, we can derive many formulas for the convolution sums i6jn σiσj and
i12jn σiσj, and here we just list two of them.
Theorem 4.1. Let n be a positive integer. One has
1 − 6n
n
1
1 n 1 − n
An
σ3 n σ3
σn σ
,
σiσ j 108
27
2
24
24
6
648
i6jn
n 1 n
7 n
11
1
σ3 n σ3
− σ3
− σ3
σiσ j 864
108
2
96
3
48
6
i12jn
4.2
7 n 2−n
6n − 1 n An Bn
σ3
σn −
σ
−
,
3
12
48
24
12
2592
128
4.3
Journal of Applied Mathematics
11
where
6 2 2 6
∞
q; q ∞ q ; q ∞
n
1 Anq 2 2 ,
q3 ; q3 ∞ q6 ; q6 ∞
n1
4.4
12 6 2
q3 ; q3 ∞ q6 ; q6 ∞ q4 ; q4 ∞
1 Bnq 4 2 6 .
q; q ∞ q2 ; q2 ∞ q12 ; q12 ∞
n1
∞
4.5
n
Remark 4.2. Alaca and Williams 11 derived the representation for
i6jn
σiσj
σ3 n σ3 n/2 3σ3 n/3 3σ3 n/6
σiσ j 120
30
40
10
i6jn
1 − 6n
n
1−n
c6 n
σn σ
,
−
24
24
6
120
4.6
where
∞
n1
2 2 2
2 q3 ; q3
q6 ; q6 .
c6 nqn q q; q ∞ q2 ; q2
∞
Remark. Alaca et al. 12 also derived the representation for
∞
∞
i12jn
4.7
σiσj
σ3 n σ3 n/2 3σ3 n/3 σ3 n/4 9σ3 n/6
σiσ j 480
160
160
30
160
i12jn
1 − 6n n 11
3σ3 n/12 2 − n
σn σ
−
c1,12 n,
10
48
24
12
480
4.8
where
2 2 2 3 3 3 4 4 3 6 6 2
∞
q ;q ∞ q ;q ∞ q ;q ∞ q ;q ∞
n
11 c1,12 nq 10q
12 12 q; q ∞ q ; q ∞
n1
8 8
q2 ; q2 ∞ q6 ; q6 ∞
q 2 2 2 2 .
q; q ∞ q3 ; q3 ∞ q4 ; q4 ∞ q12 ; q12 ∞
4.9
12
Journal of Applied Mathematics
Proof. From 3.9, we have
6 6
q2 ; q2 ∞
∞
3 3 2 6 6 2
q ;q ∞ q ;q ∞
q; q
1
∞
∞
∞
33 σ3 nqn − 24 σ3 nq2n 810 σ3 nq6n
2 n1
n1
n1
2
∞
∞
∞
45 6n
n
6n
135 σnq −
σnq − 1944
σnq
2 n1
n1
n1
648
∞
σnq
n1
6n
∞
σnq
n
∞
σnqn
− 54
n1
4.10
2
.
n1
For n ∈ N, equating the coefficients of qn on both sides of 4.10, we obtain
n
33
n
n
45
σ3 n − 24σ3
810σ3
An 135σ
− σn
2
2
6
6
2
− 1944
σiσ j 648
σiσ j − 54
σiσ j ,
i6jn
ijn/6
4.11
ijn
where An is defined by 4.4. From 4.11, utilizing the convolution sum
σn
5
n
σ3 n − σn ,
σiσ j 12
2
12
ijn
4.12
we can derive the formula 4.2.
From 3.23, we have
12 6 2
q3 ; q3 ∞ q6 ; q6 ∞ q4 ; q4 ∞
4 2 2 2 12 12 6
q; q ∞ q ; q ∞ q ; q ∞
1
∞
∞
∞
n3 qn
n3 q3n
n3 q6n
2
4
64 −
n
3n
9 n1 1 − q
3 n1 1 − q
3 n1 1 − q6n
2
∞
∞
∞
∞
n3 q12n
nq6n
nqn
nq6n
64 20 34 −
− 96
3 n1 1 − q12n
3 n1 1 − q6n
9 n1 1 − qn
1 − q6n
n1
∞
nq6n
32
1 − q6n
n1
2
∞
∞
∞
nqn
nq6n
nq12n
8 176 −
1 − qn
3 n1 1 − q6n
3 n1 1 − q12n
n1
∞
nq12n
768
12n
n1 1 − q
2
∞
nq12n
− 128
12n
n1 1 − q
∞
nqn
.
1 − qn
n1
4.13
Journal of Applied Mathematics
13
For n ∈ N, equating the coefficients of qn on both sides of 4.13, we obtain
Bn 64
64 n 20
n
34
n
n
2
4
σ3 n − σ3
σ3
− σ3
σ
σn
9
3
3
3
6
3
12
3
6
9
176 n −
σ
− 96
σiσ j 32
σiσ j
3
12
i6jn
ijn/6
4.14
8
σiσ j 768
σiσ j − 128
σiσ j ,
3 ijn
i12jn
ijn/12
where Bn is defined by 4.5. From 4.2, 4.12, and 4.14, we derive the formula 4.3.
The advantage of the formulas of Alaca et al. is that the values of c6 n and c1,12 n
are often very small. Numerical evidence suggests that |c6 n/120| < |An/648| and
|11c1,12 n/480| < |An/2592 − Bn/128|. The advantage of our formulas is that the signs
of An and An/2592 − Bn/128 appear to have periodicity. Numerical evidence suggests
that for n ≥ 3, An > 0 when 3 | n and An < 0 when, 3 n and An/2592 − Bn/128 <
0. Therefore, we conjecture that, for n ≥ 3, we have
1 − 6n
n
1
1 n 1 − n
σ3 n σ3
σn σ
,
σiσ j >
108
27
2
24
24
6
i6jn
1
1 n 1 − n
σ3 n σ3
σn,
σiσ j <
108
27
2
24
i6jn
3 | n,
4.15
3 n.
Acknowledgments
The authors would like to thank the anonymous referee very much for valuable suggestions,
corrections, and comments which resulted in a great improvement of the original paper. This
work was supported by the National Natural Science Foundation of China and the Jiangsu
University Foundation Grants 11JDG035 and 11JDG036.
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