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International Journal of Number Theory
Vol. 12, No. 4 (2016) 945–954
c World Scientific Publishing Company
DOI: 10.1142/S1793042116500585
On the number of representations of
integers by sums of mixed numbers
Int. J. Number Theory 2016.12:945-954. Downloaded from www.worldscientific.com
by SOUTHEAST UNIVERSITY on 09/11/16. For personal use only.
Ernest X. W. Xia∗ and Y. H. Ma†
Department of Mathematics
Jiangsu University
Zhenjiang, Jiangsu 212013, P. R. China
∗[email protected]
†[email protected]
L. X. Tian
School of Mathematical Sciences
Nanjing Normal University
Nanjing 210023, P. R. China
[email protected]
Received 23 August 2014
Accepted 6 July 2015
Published 16 October 2015
In this paper, several explicit formulas for the number of representations of a positive
integer by sums of mixed numbers are determined by employing theta function identities
and the (p, k)-parametrization of theta functions due to Alaca, Alaca and Williams. It
is interesting that the formulas proved in this paper are linear combinations of σ3 (n),
σ3 (n/2), σ3 (n/3), σ3 (n/4), σ3 (n/6) and σ3 (n/12).
Keywords: Sum of mixed numbers; squares; triangular numbers.
Mathematics Subject Classification 2010: 11E25, 11E20
1. Introduction and Notation
The aim of this paper is to determine explicit formulas for the number of representations of a positive integer by sums of mixed numbers by utilizing the (p, k)parametrization of theta functions due to Alaca, Alaca and Williams [1, 2, 7].
Let N0 , N and Z denote the sets of non-negative integers, positive integers and
integers, respectively. For i, n ∈ N, let
di ,
(1.1)
σi (n) =
d|n
945
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where d runs through the positive divisors of n. If n is not a positive integer, set
σi (n) = 0.
Alaca, Alaca, Williams [2], Chan and Cooper [10], Köklüce [11–13], Lomadze
[14], Xia [16], and Xia, Yao and Zhao [18] established explicit formulas for the
number of representations of a positive integer n by sums of the quadratic forms
x21 + x1 x2 + x22 . Recently, Alaca, Alaca and Williams [4, 6, 8] proved a number
of formulas for the number of representations of a positive integer n by sums of
squares. In this paper, we consider the number of representations of integers by
sums of mixed numbers. Let N (a1 , a2 , a3 ; b1 , b2 ; c1 , c2 , c3 , c4 ; n) denote the number
of representations of n by the form
a1
(wi2 + wi mi + m2i ) + 2
i=1
a2
(u2i + ui vi + vi2 )
i=1
+4
a3
(ri2 + ri si + s2i ) +
i=1
+3
p2i + 3
i=1
c3
y 2 + yi
i
i=1
b1
2
+3
c4
b2
qi2 +
i=1
c1 2
t + ti
i
i=1
2
+
c2
(x2i + xi )
i=1
(zi2 + zi ),
i=1
where wi , mi , ui , vi , ri , si , pi , qi ∈ Z and ti , xi , yi , zi ∈ N0 . Alaca, Alaca,
Lemire and Williams [1] established some theta function identities and used those
identities to derive formulas for N (1, 0, 0; 2, 0; 0, 0, 0, 0; n), N (0, 1, 0; 2, 0; 0, 0, 0, 0; n),
N (0, 0, 1; 2, 0; 0, 0, 0, 0; n), N (1, 0, 0; 0, 2; 0, 0, 0, 0; n), N (0, 1, 0; 0, 2; 0, 0, 0, 0; n) and
N (0, 0, 1; 0, 2; 0, 0, 0, 0; n). Xia and Yao [17] established some formulas for
N (a1 , a2 , a3 ; b1 , b2 ; c1 , c2 , c3 , c4 ; n) for some values of ai , bi and ci , but those formulas
are different from the formulas proved in this paper. In this paper, we prove several
formulas for N (a1 , a2 , a3 ; b1 , b2 ; c1 , c2 , c3 , c4 ; n). It is interesting that our formulas
for N (a1 , a2 , a3 ; b1 , b2 ; c1 , c2 , c3 , c4 ; n) stated in this paper are linear combinations
of σ3 (n), σ3 (n/2), σ3 (n/3), σ3 (n/4), σ3 (n/6) and σ3 (n/12).
The main results of this paper can be stated as follows.
Theorem 1.1. For n ∈ N, we have
N (0, 1, 0; 3, 3; 0, 0, 0, 0; 2n) = N (1, 1, 1; 1, 1; 0, 0, 0, 0; 2n)
= 18σ3 (n) − 48σ3 (n/2)
− 162σ3(n/3) + 432σ3 (n/6),
(1.2)
N (0, 1, 0; 2, 2; 0, 1, 0, 1; 2n − 1) = 4σ3 (n) − 4σ3 (n/2)
− 36σ3 (n/3) + 36σ3 (n/6),
N (0, 1, 0; 1, 1; 0, 2, 0, 2; 2n − 2) = N (0, 0, 1; 2, 0; 1, 0, 1, 2; 2n − 2)
= N (0, 0, 1; 0, 2; 1, 2, 1, 0; 2n − 1)
(1.3)
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947
= N (1, 1, 0; 0, 0; 0, 2, 0, 2; 2n − 2)
= σ3 (n) − σ3 (n/2) − 9σ3 (n/3) + 9σ3 (n/6),
(1.4)
N (1, 1, 0; 0, 0; 2, 0, 2, 0; n − 1) = σ3 (n) − σ3 (n/2)
− 9σ3 (n/3) + 9σ3 (n/6),
N (1, 1, 0; 0, 4; 0, 0, 0, 0; 2n) = 6σ3 (n) + 234σ3 (n/3),
(1.5)
(1.6)
N (1, 1, 0; 2, 2; 0, 0, 0, 0; 2n) = 34σ3 (n) − 64σ3 (n/2)
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− 306σ3 (n/3) + 576σ3 (n/6),
N (1, 1, 0; 4, 0; 0, 0, 0, 0; 2n) = 78σ3 (n) + 162σ3 (n/3),
(1.7)
(1.8)
N (0, 1, 1; 2, 2; 0, 0, 0, 0; 2n) = 10σ3 (n) − 40σ3 (n/2)
− 90σ3 (n/3) + 360σ3 (n/6),
(1.9)
N (0, 1, 1; 0, 0; 2, 0, 2, 0; 2n − 1) = 2σ3 (n) − 2σ3 (n/2)
− 18σ3 (n/3) + 18σ3 (n/6),
(1.10)
N (0, 3, 0; 1, 1; 0, 0, 0, 0; n) = 2σ3 (n) − 18σ3 (n/3)
− 32σ3 (n/4) + 288σ3 (n/12),
N (3, 0, 0; 0, 0; 1, 0, 1, 0; n) = σ3 (2n + 1) − 9σ3 ((2n + 1)/3),
N (0, 0, 3; 1, 1; 0, 0, 0, 0; 2n) = 24σ3 (n/2) + 216σ3 (n/6),
(1.11)
(1.12)
(1.13)
N (2, 1, 0; 1, 1; 0, 0, 0, 0; 2n) = 66σ3 (n) − 96σ3 (n/2)
− 594σ3 (n/3) + 864σ3 (n/6),
(1.14)
N (2, 1, 0; 0, 0; 0, 1, 0, 1; 2n − 1) = 12σ3 (n) − 12σ3 (n/2)
− 108σ3 (n/3) + 108σ3 (n/6),
N (1, 2, 0; 1, 1; 0, 0, 0, 0; 2n) = 24σ3 (n) + 216σ3 (n/3),
(1.15)
(1.16)
N (0, 1, 2; 1, 1; 0, 0, 0, 0; 2n) = 6σ3 (n) − 36σ3 (n/2)
− 54σ3 (n/3) + 324σ3 (n/6),
(1.17)
N (1, 1, 1; 0, 0; 0, 1, 0, 1; 2n − 1) = 6σ3 (n) − 6σ3 (n/2)
− 54σ3 (n/3) + 54σ3 (n/6).
(1.18)
It should be noted that all of the results in Theorem 1.1 involve weight 4 modular
forms, and so
1
a1 + a2 + a3 + (b1 + b2 + c1 + c2 + c3 + c4 ) = 4.
2
From Theorem 1.1, we can obtain some interesting identities. For example, by (1.3)
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948
and (1.18), we see that for n ∈ N,
3N (0, 1, 0; 2, 2; 0, 1, 0, 1; 2n − 1) = 2N (1, 1, 1; 0, 0; 0, 1, 0, 1; 2n − 1).
(1.19)
In should be noted that in general,
3N (0, 1, 0; 2, 2; 0, 1, 0, 1; 2n) = 2N (1, 1, 1; 0, 0; 0, 1, 0, 1; 2n).
(1.20)
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2. The (p, k)-Parametrization of Theta Functions
In this section, we recall some basic facts about the (p, k)-parametrization of Eisenstein series and theta functions due to Alaca, Alaca and Williams [1–3, 7].
In his second notebook [15], Ramanujan gave the definitions of three Eisenstein
series and one of them is
∞
n3 q n
.
(2.1)
M (q) = 1 + 240
1 − qn
n=1
It is trivial to check that
∞
σ3 (n)q n .
(2.2)
p = p(q) =
ϕ2 (q) − ϕ2 (q 3 )
,
2ϕ2 (q 3 )
(2.3)
k = k(q) =
ϕ3 (q 3 )
,
ϕ(q)
(2.4)
M (q) = 1 + 240
n=1
Following [2], we set
where ϕ(q) is defined by
ϕ(q) =
∞
2
qn .
(2.5)
n=−∞
Alaca and Williams [7] derived the representations of M (q), M (q 2 ), M (q 3 ) and
M (q 6 ) in terms of p and k. Equations (3.69)–(3.72) in [7] are
M (q) = (1 + 124p + 964p2 + 2788p3 + 3910p4
+ 2788p5 + 964p6 + 124p7 + p8 )k 4 ,
(2.6)
M (q 2 ) = (1 + 4p + 64p2 + 178p3 + 235p4 + 178p5 + 64p6 + 4p7 + p8 )k 4 ,
(2.7)
M (q 3 ) = (1 + 4p + 4p2 + 28p3 + 70p4 + 28p5 + 4p6 + 4p7 + p8 )k 4
(2.8)
and
M (q 6 ) = (1 + 4p + 4p2 − 2p3 − 5p4 − 2p5 + 4p6 + 4p7 + p8 )k 4 ,
(2.9)
respectively. Alaca, Alaca and Williams [3] also deduced the representations of
M (q 4 ) and M (q 12 ) in terms of p and k. Equations (3.17) and (3.19) in [3] are
31 6 29 7
1 8
4
2
3
4
5
M (q ) = 1 + 4p + 4p − 2p + 10p + 28p + p − p + p k 4 (2.10)
4
4
16
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On the number of representations of integers by sums of mixed numbers
and
1 6 1 7
1 8
2
3
4
5
M (q ) = 1 + 4p + 4p − 2p − 5p − 2p + p + p + p k 4 ,
4
4
16
12
949
(2.11)
respectively. Jonathan and Peter Borwein [9] introduced three 2-dimensional theta
functions and one of them is
∞
∞
2
2
a(q) =
q i +ij+j .
(2.12)
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i=−∞ j=−∞
Alaca, Alaca and Williams [2] found the representations of a(q), a(q 2 ) and a(q 4 ) in
terms of p and k. From [2, Theorems 1, 2 and 4], we have
a(q) = (1 + 4p + p2 )k,
2
2
a(q ) = (1 + p + p )k,
1
a(q 4 ) = 1 + p − p2 k.
2
(2.13)
(2.14)
(2.15)
Alaca, Alaca, Lemire and Williams [1] gave the representations of ϕ(q) and ϕ(q 3 )
in terms of p and k. From [1, (2.3)], we obtain
3
1
1
1
ϕ(q) = (1 + 2p) 4 k 2
(2.16)
and
ϕ(q 3 ) = (1 + 2p) 4 k 2 .
(2.17)
3. Several Theta Function Identities
In order to prove the main results of this paper, we establish several theta function
identities in this section. Define
∞
∞
(1 − q 4n )6 (1 − q 6n )4
h(n)q n = q
,
(3.1)
H(q) =
(1 − q 12n )2
n=0
n=1
G(q) =
∞
g(n)q n = q 3
n=0
∞
(1 − q 2n )4 (1 − q 12n )6
(1 − q 4n )2
n=1
(3.2)
and
ψ(q) =
∞
q
n2 +n
2
.
n=0
Theorem 3.1. The following identities hold:
f = k1 M (q) + k2 M (q 2 ) + k3 M (q 3 ) + k4 M (q 4 )
+ k6 M (q 6 ) + k12 M (q 12 ) + kG G(q) + kH H(q),
where
(3.3)
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E. X. W. Xia, Y. H. Ma & L. X. Tian
f
2
k1
3
3
3
a(q )ϕ (q)ϕ (q )
2
2
2
3
2
6
qa(q )ϕ (q)ϕ (q )ψ(q )ψ(q )
q 2 a(q 2 )ϕ(q)ϕ(q 3 )ψ 2 (q 2 )ψ 2 (q 6 )
2
4
2
3
2
6
q a(q )ϕ (q)ψ(q)ψ(q )ψ (q )
4
2
3
2
2
3
qa(q )ϕ (q )ψ(q)ψ (q )ψ(q )
2
4
3
a(q)a(q )ϕ (q )
a(q)a(q 2 )ϕ2 (q)ϕ2 (q 3 )
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203-IJNT
a(q)a(q 2 )ϕ4 (q)
qa(q)a(q 2 )ψ 2 (q)ψ 2 (q 3 )
2
2
2
2
2
6
q a(q)a(q )ψ (q )ψ (q )
a(q 2 )a(q 4 )ϕ2 (q)ϕ2 (q 3 )
2
4
2
2
3
qa(q )a(q )ψ (q)ψ (q )
a3 (q 2 )ϕ(q)ϕ(q 3 )
3
2
3
4
2
6
qa (q )ψ(q )ψ(q )
3
a (q )ϕ(q)ϕ(q )
a2 (q)a(q 2 )ϕ(q)ϕ(q 3 )
1
120
1
480
1
1920
1
1920
1
1920
1
300
1
60
13
300
1
240
1
640
1
240
1
960
1
120
1
240
1
1200
1
30
k2
k3
k4
2
− 15
3
− 80
3
− 400
1
− 40
3
− 40
3
− 160
3
− 640
3
− 640
3
− 640
13
100
3
− 20
9
100
3
− 80
9
− 640
3
− 80
3
− 320
3
− 40
3
− 80
3
400
3
− 10
1
30
8
75
2
− 15
27
80
27
− 400
9
40
6
5
3
− 10
24
25
6
5
0
1
− 480
1
− 1920
1
− 1920
1
− 1920
1
− 200
1
− 120
13
− 200
1
− 240
19
− 1920
1
240
1
− 960
0
0
0
0
0
2
75
2
− 15
26
75
0
1
120
2
− 15
0
2
− 15
k6
k12
kG
kH
0
6
5
12
4
3
2
− 38
21
8
− 38
− 18
5
1
2
− 81
− 81
7
8
26
5
3
160
3
640
3
640
3
640
39
− 200
3
40
27
− 200
3
80
57
640
3
− 80
3
320
0
0
0
0
0
26
25
6
5
18
25
18
6
− 234
5
18
5
0
0
0
3
− 40
6
5
− 98
− 83
9
3
9
4
3
4
0
0
0
0
− 27
5
9
5
18
6
0
qa2 (q)a(q 2 )ψ(q 2 )ψ(q 6 )
1
96
7
− 160
3
− 32
1
30
63
160
3
− 10
− 92
− 23
a(q)a2 (q 2 )ϕ(q)ϕ(q 3 )
1
75
1
− 50
3
25
8
75
9
− 50
24
25
− 72
5
24
5
a(q 2 )a2 (q 4 )ϕ(q)ϕ(q 3 )
1
480
1
160
3
2
9
− 160
− 15
− 160
6
5
9
2
3
2
a(q)a(q 2 )a(q 4 )ϕ(q)ϕ(q 3 )
1
120
0
3
− 40
0
6
5
18
6
qa(q)a(q 2 )a(q 4 )ψ(q 2 )ψ(q 6 )
1
960
1
64
3
1
− 320
− 60
9
− 64
3
20
9
4
3
4
2
− 15
Proof. We just prove the last identity, that is,
qa(q)a(q 2 )a(q 4 )ψ(q 2 )ψ(q 6 ) =
1
1
3
1
M (q) + M (q 2 ) −
M (q 3 ) − M (q 4 )
960
64
320
60
9
9
3
3
− M (q 6 ) + M (q 12 ) + H(q) + G(q). (3.4)
64
20
4
4
The rest can be proved similarly. By the well-known Jacobi triple product identity,
ψ(q) =
where ψ(q) is defined by (3.3).
∞
(1 − q 2n )2
,
(1 − q n )
n=1
(3.5)
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951
j ∞
Alaca, Alaca and Williams [5] established the representations of q 24 n=1 (1 −
q nj ) (j = 1, 2, 3, 4, 6, 12) in terms of p and k. It follows from [5, (2.10)–(2.15)] that
1
q 24
∞
1
1
1
1
1
1
1
1
1
1
(1 − q n ) = 2− 6 p 24 (1 − p) 2 (1 + p) 6 (1 + 2p) 8 (2 + p) 8 k 2 ,
(3.6)
n=1
1
q 12
∞
1
1
1
1
(1 − q 2n ) = 2− 3 p 12 (1 − p) 4 (1 + p) 12 (1 + 2p) 4 (2 + p) 4 k 2 ,
(3.7)
n=1
1
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q8
∞
1
1
1
1
1
1
1
(1 − q 3n ) = 2− 6 p 8 (1 − p) 6 (1 + p) 2 (1 + 2p) 24 (2 + p) 24 k 2 ,
(3.8)
n=1
1
q6
∞
1
1
1
1
1
1
1
1
(1 − q 4n ) = 2−2/3 p 6 (1 − p) 8 (1 + p) 24 (1 + 2p) 8 (2 + p) 2 k 2 ,
(3.9)
n=1
1
q4
∞
1
1
1
1
1
(1 − q 6n ) = 2− 3 p 4 (1 − p) 12 (1 + p) 4 (1 + 2p) 12 (2 + p) 12 k 2 ,
(3.10)
n=1
q
1
2
∞
1
1
1
1
1
1
(1 − q 12n ) = 2−2/3 p 2 (1 − p) 24 (1 + p) 8 (1 + 2p) 24 (2 + p) 6 k 2 . (3.11)
n=1
In view of (3.7) and (3.9)–(3.11),
H(q) =
p(1 − p)(1 + p)(1 + 2p)(2 + p)3 4
k
16
(3.12)
G(q) =
p3 (1 − p)(1 + p)(1 + 2p)(2 + p) 4
k ,
16
(3.13)
and
where H(q) and G(q) are defined by (3.1) and (3.2), respectively. Thanks to (2.13)–
(2.15), (3.5), (3.7) and (3.9)–(3.11),
qa(q)a(q 2 )a(q 4 )ψ(q 2 )ψ(q 6 )
1
13 2 27 3 55 4 29 5 5 7 1 8
p + p + p + p + p − p − p k4 .
=
2
4
4
8
8
8
8
(3.14)
By (2.6)–(2.11), (3.12) and (3.13),
1
1
3
1
9
M (q) + M (q 2 ) −
M (q 3 ) − M (q 4 ) − M (q 6 )
960
64
320
60
64
9
3
3
+ M (q 12 ) + H(q) + G(q)
20
4
4
1
13 2 27 3 55 4 29 5 5 7 1 8
=
p + p + p + p + p − p − p k4 .
2
4
4
8
8
8
8
Identity (3.4) follows from (3.14) and (3.15). This completes the proof.
(3.15)
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4. Proof of Theorem 1.1
In this section, we present a proof of Theorem 1.1 by utilizing Theorem 3.1. We
deduce (1.18) from (3.4). The rest can be proved similarly. It is easy to see that
1+
∞
N (a1 , a2 , a3 ; b1 , b2 ; c1 , c2 , c3 , c4 ; n)q n
n=1
= aa1 (q)aa2 (q 2 )aa3 (q 4 )ϕb1 (q)ϕb2 (q 3 )ψ c1 (q)ψ c2 (q 2 )ψ c3 (q 3 )ψ c4 (q 6 ).
(4.1)
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Setting a1 = a2 = a3 = c2 = c4 = 1 and b1 = b2 = c1 = c3 = 0 in (4.1) and
multiplying q on both sides, and then employing (2.2), (3.1), (3.2), (3.4), we get
q+
∞
N (1, 1, 1; 0, 0; 0, 1, 0, 1; n)q n+1
n=1
= qa(q)a(q 2 )a(q 4 )ψ(q 2 )ψ(q 6 )
∞
∞
1
1
n
2n
1 + 240
1 + 240
σ3 (n)q
σ3 (n)q
+
=
960
64
n=0
n=0
∞
∞
3
1
−
σ3 (n)q 3n −
σ3 (n)q 4n
1 + 240
1 + 240
320
60
n=0
n=0
∞
∞
9
3
6n
12n
−
1 + 240
1 + 240
σ3 (n)q
σ3 (n)q
+
64
20
n=0
n=0
+
=
∞
∞
3
9
h(n)q n +
g(n)q n
4 n=1
4 n=1
∞
∞
∞
∞
1
15 9
σ3 (n)q n +
σ3 (n)q 2n −
σ3 (n)q 3n − 4
σ3 (n)q 4n
4 n=0
4 n=0
4 n=0
n=0
−
∞
∞
∞
∞
135 3
9
σ3 (n)q 6n + 36
σ3 (n)q 12n +
h(n)q n +
g(n)q n .
4 n=0
4
4
n=0
n=1
n=1
(4.2)
Equating the coefficients of q 2n on both sides of (4.2), we find that for n ∈ N,
N (1, 1, 1; 0, 0; 0, 1, 0, 1; 2n − 1) =
15
9
1
σ3 (2n) + σ3 (n) − σ3 (2n/3) − 4σ3 (n/2)
4
4
4
135
9
3
−
σ3 (n/3) + 36σ3 (n/6) + h(2n) + g(2n).
4
4
4
(4.3)
It is easy to check that for n ∈ N,
σ3 (2n) = 9σ3 (n) − 8σ3 (n/2)
(4.4)
March 25, 2016 9:10 WSPC/S1793-0421
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On the number of representations of integers by sums of mixed numbers
953
and
σ3 (2n/3) = 9σ3 (n/3) − 8σ3 (n/6).
(4.5)
By (3.1) and (3.2), we see that for n ∈ N,
h(2n) = g(2n) = 0.
(4.6)
Identity (1.18) follows from (4.3)–(4.6). The proof is complete.
Int. J. Number Theory 2016.12:945-954. Downloaded from www.worldscientific.com
by SOUTHEAST UNIVERSITY on 09/11/16. For personal use only.
Acknowledgments
This work was supported by the National Science Foundation of China (11571043)
and CPSF (2014M551506, 2015T80499).
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