MATEMATIQKI VESNIK
originalni nauqni rad
research paper
66, 3 (2014), 283–293
September 2014
ON SOME NEW MIXED MODULAR EQUATIONS
INVOLVING RAMANUJAN’S THETA-FUNCTIONS
M. S. Mahadeva Naika, S. Chandankumar and M. Harish
Abstract. In his second notebook, Ramanujan recorded altogether 23 P –Q modular equations involving his theta functions. In this paper, we establish several new mixed modular equations involving Ramanujan’s theta-functions ϕ and ψ which are akin to those recorded in his
notebook.
1. Introduction
Following Ramanujan, let (a; q)∞ denote the infinite product
q are complex numbers, |q| < 1) and
f (a, b) :=
∞
P
∞
Q
(1 − aq n ) (a,
n=0
an(n+1)/2 bn(n−1)/2 , |ab| < 1,
n=−∞
denote the Ramanujan theta-function. The product representation of f (a, b) follows
from Jacobi’s triple product identity and is given by:
f (a, b) = (−a; ab)∞ (−b; ab)∞ (ab; ab)∞ .
The following definitions of theta-functions ϕ, ψ and f with |q| < 1 are classical:
∞
P
ϕ(q) := f (q, q) =
2
n=−∞
ψ(q) := f (q, q 3 ) =
∞
P
q n = (−q; q 2 )2∞ (q 2 ; q 2 )∞ ,
q n(n+1)/2 =
n=0
f (−q) := f (−q, −q 2 ) =
∞
P
n=−∞
(q 2 ; q 2 )∞
,
(q; q 2 )∞
(−1)n q n(3n−1)/2 = (q; q)∞ .
The ordinary or Gaussian hypergeometric function is defined by
2 F1 (a, b; c; z)
:=
∞ (a) (b)
P
n
n n
z ,
n=0 (c)n n!
0 ≤ |z| < 1,
2010 Mathematics Subject Classification: 33D10, 11A55, 11F27
Keywords and phrases: Modular equations; theta-functions
283
(1.1)
(1.2)
(1.3)
284
M.S. Mahadeva Naika, S. Chandankumar, M. Harish
where a, b, c are complex numbers such that c 6= 0, −1, −2, . . . , and
(a)0 = 1, (a)n = a(a + 1) · · · (a + n − 1) for any positive integer n.
Now we recall the notion of modular equation. Let K(k) be the complete elliptic
integral of the first kind of modulus k. Recall that
¡ 1 ¢2
Z π/2
∞
dφ
π P
π
2 n 2n
p
K(k) :=
=
k = ϕ2 (q), (0 < k < 1), (1.4)
2 n=0 (n!)2
2
0
1 − k 2 sin2 φ
√
and set K 0 = K(k 0 ), where k 0 = 1 − k 2 is the so called complementary modulus
0
of k. It is classical to set q(k) = e−πK(k )/K(k) so that q is one-to-one, increasing
from 0 to 1.
In the same manner introduce L1 = K(`1 ), L01 = K(`01 ) and suppose that the
following equality
K0
L0
n1
= 1
(1.5)
K
L1
holds for some positive integer n1 . Then a modular equation of degree n1 is an
algebraic relation between the moduli k and `1 which is induced by (1.5). Following
Ramanujan, set α = k 2 and β = `21 . Then we say that β is of degree n1 over α.
The multiplier m is defined by
m=
K
ϕ2 (q)
= 2 n1 ,
L1
ϕ (q )
(1.6)
0
for q = e−πK(k )/K(k) .
of the
Let K, K 0 , L1 , L01 , L2 , L02 , L3 and L03 denote complete elliptic integrals
√
√ √ √
first kind corresponding, in pairs, to the moduli α, β, γ and δ, and their
complementary moduli, respectively. Let n1 , n2 and n3 be positive integers such
that n3 = n1 n2 . Suppose that the equalities
n1
K0
L0
K0
L0
K0
L0
= 1 , n2
= 2 and n3
= 3
K
L1
K
L2
K
L3
(1.7)
√ √
hold. Then
√ a “mixed” modular equation is a relation between the moduli α, β,
√
γ and δ that is induced by (1.7). We say that β, γ and δ are of degrees n1 ,
n2 and n3 respectively over α. The multipliers m = K/L1 and m0 = L2 /L3 are
algebraic relation involving α, β, γ and δ.
At scattered places of his second notebook [5], Ramanujan recorded a total of
nine P –Q mixed modular relations of degrees 1, 3, 5 and 15. For example, if
P :=
f (−q 3 )f (−q 5 )
q 1/3 f (−q)f (−q 15 )
then
and Q :=
f (−q 6 )f (−q 10 )
q 2/3 f (−q 2 )f (−q 30 )
,
³ Q ´3 ³ P ´3
1
=
+
+ 4.
(1.8)
PQ
P
Q
The proofs of these nine P –Q mixed modular relations can be found in the book
by Berndt [2, pp. 214–235]. In [3,4], S. Bhargava, C. Adiga and M. S. Mahadeva
PQ +
Mixed modular equations involving Ramanujan’s theta-functions
285
Naika have established several new P –Q mixed modular relations with four moduli. These mixed modular equations are employed to evaluate explicit evaluations
of Ramanujan-Weber class invariants, ratios of Ramanujan’s theta functions and
several continued fractions.
Motivated by all these works, in this article, we establish several new mixed
modular equations involving Ramanujan’s theta-functions.
In Section 2, we collect some identities which are useful in proofs of our main
results. In Section 3, we establish several new P –Q mixed modular equations akin
to those recorded by Ramanujan in his notebooks involving his theta-functions ϕ(q)
and ψ(q).
2. Preliminary results
In this section, we collect some identities which are useful in establishing our
main results.
Lemma 2.1. [1, Ch. 16, Entry 24 (ii), p. 39] We have
f 3 (−q) = ϕ2 (−q)ψ(q).
(2.1)
Lemma 2.2. [1, Ch. 16, Entry 24 (iv), p. 39] We have
f 3 (−q 2 ) = ϕ(−q)ψ 2 (q).
(2.2)
Lemma 2.3. [1, Ch. 17, Entry 10 (i) and Entry 11(ii), pp. 122–123] For
0 < α < 1, we have
√
ϕ(q) = z,
(2.3)
√ 1/8
√
1/8
2q ψ(−q) = z{α(1 − α)} .
(2.4)
Lemma 2.4. [1, Ch. 20, Entry 3 (xii), (xiii), pp. 352–353] Let α, β and γ be of
the first, third and ninth degrees respectively. Let m denote the multiplier relating
α, β and m0 be the multiplier relating γ, δ. Then
³ β 2 ´1/4 ³ (1 − β)2 ´1/4 ³
´1/4
β 2 (1 − β)2
m
+
−
= −3 0 ,
αγ
(1 − α)(1 − γ)
αγ(1 − α)(1 − γ)
m
(2.5)
³ αγ ´1/4 ³ (1 − α)(1 − γ) ´1/4 ³ αγ(1 − α)(1 − γ) ´1/4
0
m
+
−
=
.
(2.6)
β2
(1 − β)2
β 2 (1 − β)2
m
Lemma 2.5. [1, Ch. 20, Entry 11 (viii), (ix), p. 384] Let α, β, γ and δ be of
the first, third, fifth and fifteenth degrees respectively. Let m denote the multiplier
286
M.S. Mahadeva Naika, S. Chandankumar, M. Harish
relating α and β and m0 be the multiplier relating γ and δ. Then
³ αδ ´1/8 ³ (1 − α)(1 − δ) ´1/8 ³ αδ(1 − α)(1 − δ) ´1/8 r m0
+
−
=
,
βγ
(1 − β)(1 − γ)
βγ(1 − β)(1 − γ)
m
(2.7)
r
³ βγ ´1/8 ³ (1 − β)(1 − γ) ´1/8 ³ βγ(1 − β)(1 − γ) ´1/8
m
+
−
=−
.
αδ
(1 − α)(1 − δ)
αδ(1 − α)(1 − δ)
m0
(2.8)
q 1/6 f (−q)f (−q 9 )
q 1/3 f (−q 2 )f (−q 18 )
and
M
:=
, then
f 2 (−q 3 )
f 2 (−q 6 )
h L3
L9
M9
M3 i
1
+ 9 + 12
+ 3 + 27L3 M 3 − 3 3 = 0.
(2.9)
9
3
M
L
M
L
L M
Lemma 2.6. [3] If L :=
Lemma 2.7. [2,3] If L :=
then
L3 M 3 +
1
L3 M 3
q 2/3 f (−q 2 )f (−q 30 )
q 1/3 f (−q)f (−q 15 )
and
M
:=
,
f (−q 3 )f (−q 5 )
f (−q 6 )f (−q 10 )
h L3
h L6
M3 i
M 6 i h L9
M9 i
+ 3 − 12
+ 6 −
+ 9 = 76. (2.10)
3
6
9
M
L
M
L
M
L
− 48
3. New mixed modular equations
In this section, we establish several new mixed modular equations involving
Ramanujan’s theta-functions ϕ(q) and ψ(q).
√
qψ(−q)ψ(−q 9 )
ϕ(q)ϕ(q 9 )
Theorem 3.1. If L :=
and
M
:=
, then
ϕ2 (q 3 )
ψ 2 (−q 3 )
L2 =
M2 + 1
.
1 − 3M 2
Proof. The equations (2.5) and (2.6) can be rewritten as
m ³ αγ(1 − α)(1 − γ) ´1/4
m0 ³ αγ(1 − α)(1 − γ) ´1/4
+
=
1
−
3
.
m
β 2 (1 − β)2
m0
β 2 (1 − β)2
(3.1)
(3.2)
Transforming the above equation (3.2) by using the equations (2.3) and (2.4), we
arrive at the equation (3.1).
√
qψ(−q)ψ(−q 9 )
qψ(−q 2 )ψ(−q 18 )
Theorem 3.2. If P :=
and
Q
:=
, then
ψ 2 (−q 3 )
ψ 2 (−q 6 )
P2
Q2 ³ P 2
Q2 ´2
+ 2+
+ 2
2
2
Q
P
Q
P
³
1
3P
3Q P 3
Q3 ´
1 ´³
3P Q −
+
+
+ 3 + 3 + 14. (3.3)
= 3P Q −
PQ
PQ
Q
P
Q
P
Mixed modular equations involving Ramanujan’s theta-functions
287
Proof. Using the equation (2.2) in the equation (2.9), we deduce
a3 P 12 − aP 4 bQ4 + 27a2 P 8 b2 Q8 + 12a2 P 8 bQ4 + 12aP 4 b2 Q8 + b3 Q12 = 0,
2
2
(3.4)
9
ϕ (q)ϕ (q )
and b := a(q 2 ).
ϕ4 (q 3 )
Using the equation (3.1), we have
where a := a(q) :=
a :=
P2 + 1
Q2 + 1
and b :=
.
2
1 − 3P
1 − 3Q2
(3.5)
Employing the equations (3.5) in the equation (3.4), we find
A(P, Q)B(P, Q) = 0,
(3.6)
where
¡
A(P, Q) := − 2P 2 Q4 − 13P 6 Q2 + 25P 6 Q4 + 9P 4 Q4 − 13P 2 Q6 − 16P 8 Q2
+ 39P 8 Q4 − 6P 10 Q2 − 27P 6 Q6 − 18P 8 Q6 + 25P 4 Q6 − 16P 2 Q8 + 39P 4 Q8
+ 9P 10 Q4 − 18P 6 Q8 − 27P 8 Q8 − 6P 2 Q10 + 9P 4 Q10 + P 2 Q2 + P 6 + Q6
¢
+ P 10 + 2P 8 + 2Q8 + Q10 − 2P 4 Q2
and
¡
B(P, Q) := − P 8 + 3P 8 Q2 − P 6 Q2 − P 6 + 9P 6 Q4 + 9P 6 Q6 − Q6 + 9P 4 Q6
¢
+ 6P 4 Q4 − 3P 4 Q2 + 3P 2 Q8 − P 2 Q6 − 3P 2 Q4 + P 2 Q2 − Q8 .
Expanding in powers of q, the first and second factor of the equation (3.6), one gets
respectively,
¡
¢
A(P, Q) = −q 3 − 1 − q + q 2 + 4q 3 − 10q 4 − 4q 5 + 37q 6 − 28q 7 − 48q 8 + · · ·
and
¡
¢
B(P, Q) = q 5 4 − 12q − 28q 2 + 144q 3 − 32q 4 − 764q 5 + 1424q 6 + 1400q 7 + · · · .
As q → 0, the factor q −3 B(P, Q) of the equation (3.6) vanishes whereas the other
factor q −3 A(P, Q) do not vanish. Hence, we arrive at the equation (3.3) for q ∈
(0, 1). By analytic continuation the equation (3.3) is true for |q| < 1.
√
qψ(q)ψ(q 9 )
qψ(q 2 )ψ(q 18 )
Theorem 3.3. If P :=
and
Q
:=
, then
ψ 2 (q 3 )
ψ 2 (q 6 )
Q2
1
P2
+ 2 = 4 − 3P 2 + 2 .
2
Q
P
P
(3.7)
Proof. Using the equation (2.1) in the equation (2.9), we deduce
27a2 P 4 b2 Q4 + a3 P 6 + 12a2 P 4 bQ2 + 12aP 2 b2 Q4 + b3 Q6 = aP 2 bQ2 ,
where a := a(q) =
ϕ2 (−q)ϕ2 (−q 9 )
and b := a(q 2 ).
ϕ4 (−q 3 )
(3.8)
288
M.S. Mahadeva Naika, S. Chandankumar, M. Harish
Using the equation (3.1) after changing q by −q, we find
a=
1 − P2
1 − Q2
and
b
=
.
1 + 3P 2
1 + 3Q2
(3.9)
Employing the equations (3.9) in the equation (3.8), we find
¡ 2 2
¢¡
¢
3P Q + P 2 − 1 + Q2 3P 4 Q2 + P 4 − 4P 2 Q2 − Q2 + Q4
¡
1 − 2P 2 − 2Q2 + P 4 + 44P 2 Q2 + 6P 4 Q2 + 9P 4 Q4 + 16P Q
´
+ 48P 3 Q3 + 6P 2 Q4 + Q4 = 0.
(3.10)
Expanding in powers of q, factors of the equation (3.10) becomes respectively,
¡
¢
− 1 + q + 3q 2 + 2q 3 + 2q 4 − 2q 5 − q 6 − 8q 7 − 24q 8 + · · · ,
¡
¢
q 4 − 4 − 8q − 8q 2 + 12q 3 + 52q 4 + 24q 5 − 76q 6 − 120q 7 + 32q 8 + · · ·
and
¡
¢
1 − 2q − 5q 2 + 50q 3 + 105q 4 + 68q 5 − 82q 6 − 374q 7 − 362q 8 + · · · .
As q → 0, the second factor of the equation (3.10) vanishes whereas the other
factors do not vanish. Hence, we arrive at the equation (3.7) for q ∈ (0, 1). By
analytic continuation the equation (3.7) is true for |q| < 1.
Theorem 3.4. If P :=
ϕ(−q)ϕ(−q 9 )
ϕ(−q 2 )ϕ(−q 18 )
and Q :=
, then
2
3
ϕ (−q )
ϕ2 (−q 6 )
Q2
1
P2
+ 2 = 4 − 3Q2 + 2 .
2
Q
P
Q
(3.11)
Proof. The proof of the equation (3.11) is similar to the proof of the equation
(3.7). Hence, we omit the details.
ϕ(−q)ϕ(−q 9 )
ϕ(−q 4 )ϕ(−q 36 )
and
Q
:=
, then
ϕ2 (−q 3 )
ϕ2 (−q 12 )
³P4
³P2
³P2
´
Q4 ´
Q2 ´
3Q4 ´ ³ 1
2 4
+
+
8
+
−
3
−
−
−
27P
Q
Q4
P4
Q2
P2
Q4
P2
P 2 Q4
³ 1
´
³ 1
´
³
1 ´
− 8 2 2 + 9P 2 Q2 − 12 2 − 3P 2 + 3 9Q4 + 4 = 24. (3.12)
P Q
P
Q
Theorem 3.5. If P :=
Proof. Using the equation (2.9) along with the equations (2.1) and (2.2), we
deduce
27P 4 u4 Q8 v 2 + 12P 4 u4 Q4 v + 12Q8 v 2 P 2 u2 + P 6 u6 + Q12 v 3 = P 2 u2 Q4 v,
where u :=
qψ 2 (q)ψ 2 (q 9 )
and v := u(q 4 ).
ψ 4 (q 3 )
(3.13)
Mixed modular equations involving Ramanujan’s theta-functions
289
Invoking the equation (3.1), we find
1 − P2
1 − Q2
u :=
and
v
:=
.
(3.14)
3P 2 + 1
3Q2 + 1
Using the equations (3.14) in the equation (3.13), we find
¡ 2 4
¢¡
3Q P + P 4 − 2Q2 P 2 − P 2 + 3P 2 Q4 + Q4 − Q2 3Q2 P 2 + P 2 − 1
¢¡
¢¡
+ 4QP + Q2 3Q2 P 2 + P 2 − 1 − 4QP + Q2 P 8 + 27P 6 Q8 − 72P 6 Q6
+ 36P 6 Q4 + 8P 6 Q2 − 3P 6 + 27P 4 Q8 − 24P 4 Q4 + 3P 4 + 9P 2 Q8
¢
+ 8P 2 Q6 − 12P 2 Q4 − 8Q2 P 2 − P 2 + Q8 = 0.
(3.15)
As q → 0, the last factor of the equation (3.15) vanishes whereas the other factors
do not vanish. Hence, we arrive at the equation (3.12) for q ∈ (0, 1). By analytic
continuation the equation (3.12) is true for |q| < 1.
ϕ(−q)ϕ(−q 9 )
ϕ(q)ϕ(q 9 )
and
Q
:=
, then
ϕ2 (−q 3 )
ϕ2 (q 3 )
³P
Q´ ³
1 ´
+
+ 3P Q −
− 4 = 0.
Q P
PQ
Theorem 3.6. If P :=
(3.16)
Proof. Using the equation (2.9) along with the equations (2.1) and (2.2), we
deduce
27P 8 u2 Q4 v 4 + P 12 u3 + 12P 8 u2 Q2 v 2 + 12P 4 uQ4 v 4 + Q6 v 6 = P 4 uQ2 v 2 , (3.17)
qψ 2 (q)ψ 2 (q 9 )
qψ 2 (−q)ψ 2 (−q 9 )
where u :=
and
v
:=
.
ψ 4 (q 3 )
ψ 4 (−q 3 )
Invoking the equation (3.1), we find
Q2 − 1
1 − P2
and v :=
.
(3.18)
u :=
2
3P + 1
3Q2 + 1
Using the equations (3.18) in the equation (3.17), we find
¡ 2 2
¢¡
¢¡
3P Q + Q2 − 1 + 4P Q + P 2 3P 2 Q2 + Q2 − 1 − 4P Q + P 2 3P 2 Q4
¢¡
+ Q4 − Q2 − 2P 2 Q2 − P 2 + 3P 4 Q2 + P 4 27P 8 Q6 + 27P 8 Q4 + 9P 8 Q2
+ P 8 − 72P 6 Q6 + 8P 6 Q2 + 36P 4 Q6 − 24P 4 Q4 − 12P 4 Q2 + 8P 2 Q6
¢
− 8P 2 Q2 + Q8 − 3Q6 + 3Q4 − Q2 = 0.
(3.19)
As q → 0, the second factor of the equation (3.19) vanishes whereas the other
factors do not vanish. Hence, we arrive at the equation (3.16) for q ∈ (0, 1). By
analytic continuation the equation (3.16) is true for |q| < 1.
√
qψ(q)ψ(q 9 )
q 2 ψ(q 4 )ψ(q 36 )
Theorem 3.7. If P :=
and
Q
:=
, then
ψ 2 (q 3 )
ψ 2 (q 12 )
³P4
³P2
³
³ 3P 4
Q4 ´
Q2 ´
1 ´
Q2 ´
2 2 2
+
+
8
+
−
8
3
P
Q
+
+
3
−
Q4
P4
Q2
P2
P 2 Q2
Q2
P4
´
³
´
³
´
³
1
1
1
+ 27P 4 Q2 − 4 2 + 3 9P 4 + 4 + 12 3Q2 − 2 − 24 = 0. (3.20)
P Q
P
Q
290
M.S. Mahadeva Naika, S. Chandankumar, M. Harish
Proof. The proof of the equation (3.20) is similar to the proof of the equation
(3.12). Hence, we omit the details.
√
qψ(q)ψ(q 9 )
qψ(−q)ψ(−q 9 )
and
Q
:=
, then
Theorem 3.8. If P :=
ψ 2 (q 3 )
ψ 2 (−q 3 )
³P2
Q2 ´ ³
1 ´³ P
Q´
+
−
3P
Q
+
−
+ 2 = 0.
(3.21)
Q2
P2
PQ Q P
√
Proof. The proof of the equation (3.21) is similar to the proof of the equation
(3.16). Hence, we omit the details.
Theorem 3.9. If L :=
ϕ(q 3 )ϕ(q 5 )
ψ(−q 3 )ψ(−q 5 )
and M :=
, then
15
ϕ(q)ϕ(q )
qψ(−q)ψ(−q 15 )
M=
1+L
.
1−L
Proof. The equations (2.7) and (2.8) can be rewritten as
r ³
r
m αδ(1 − α)(1 − δ) ´1/8
m
m ³ αδ(1 − α)(1 − δ) ´1/8
1+
=
−
.
m0 βγ(1 − β)(1 − γ)
m0
m0 βγ(1 − β)(1 − γ)
(3.22)
(3.23)
Transforming the above equation (3.23) by using the equations (2.3) and (2.4), we
arrive at equation (3.22).
ψ(−q 3 )ψ(−q 5 )
ψ(−q 6 )ψ(−q 10 )
and Q := 2
, then
15
qψ(−q)ψ(−q )
q ψ(−q 2 )ψ(−q 30 )
³P2
Q2 ´ ³ P
Q´ ³
1 ´
+
+
+
−
P
Q
+
Q2
P2
Q P
PQ
s
s
r
r ´
´³
³p
1
P3
Q3
P
Q
PQ − √
+
+
+
+
= 0. (3.24)
3
3
Q
P
Q
P
PQ
Theorem 3.10. If P :=
Proof. Using the equation (2.2) in the equation (2.10), we find
2
a P 4 b2 Q4 + a4 P 8 b4 Q8 − a6 P 12 − 48a4 P 8 b2 Q4 − 12a5 P 10 bQ2
− 48a2 P 4 b4 Q8 − 76a3 P 6 b3 Q6 − b6 Q12 − 12b5 Q10 aP 2 = 0,
(3.25)
ϕ(q 3 )ϕ(q 5 )
and b := a(q 2 ).
ϕ(q)ϕ(q 15 )
Using the equation (3.22), we have
where a := a(q) :=
a := a(q) :=
P −1
Q−1
and b := b(q) :=
.
P +1
Q+1
(3.26)
291
Mixed modular equations involving Ramanujan’s theta-functions
Employing the equations (3.26) in the equation (3.25), we find
¡
¢¡
¢¡
P Q + P − 1 + Q P 2 − P − 2P Q + P Q2 + Q2 P 2 Q + P 2 − Q
¢¡
− 2P Q + Q2 12P 4 Q4 − P 2 Q2 − 5P 4 Q3 − 5P 4 Q2 − 5P 3 Q4 − P 3 Q3
− P 3 Q2 − 5P 2 Q4 − P 2 Q3 − P 6 − Q6 − 3P 8 − 3Q8 + 3P 6 Q2 + 7P 5 Q2
− 3P 6 Q + 3P 2 Q6 + 7P 2 Q5 − 3P Q6 − 7P 4 Q6 − 2P 4 Q5 + 5P 3 Q6
+ 7P 3 Q5 − 12P 5 Q5 − 2P 5 Q4 + 7P 5 Q3 − 7P 6 Q4 + 5P 6 Q3 + P 6 Q6
− 5P 6 Q5 − 5P 5 Q6 + P 7 Q7 − P 7 Q6 + 5P 7 Q5 − P 6 Q7 + 5P 5 Q7 + 7P 7 Q4
+ 7P 4 Q7 + 3Q7 + Q9 + 3P 7 + P 9 + 3P 9 Q2 + P 9 Q3 − 3P 3 Q8 − 3P 3 Q7
+ P 3 Q9 + 3P 9 Q − 9P 8 Q2 + 3P 7 Q2 − 9P 8 Q + 9P 7 Q − 3P 8 Q3
− 3P 7 Q3 + 3Q9 P 2 − 9Q8 P 2 + 3Q7 P 2 + 9Q7 P − 9Q8 P + 3P Q9
¡ 4
P + Q4 − P 2 Q − P Q2 − P Q + P 4 Q − P 3 Q3 + P 3 Q2 + P 3 Q
¢
+ P 2 Q3 + P Q4 + P Q3 − P 3 − Q3 = 0.
¢
(3.27)
As q → 0, the last factor of the equation (3.27) vanishes whereas the other factors
do not vanish. Hence, we arrive at the equation (3.24) for q ∈ (0, 1). By analytic
continuation the equation (3.24) is true for |q| < 1.
ψ(q 3 )ψ(q 5 )
ψ(q 6 )ψ(q 10 )
and
Q
:=
, then
qψ(q)ψ(q 15 )
q 2 ψ(q 2 )ψ(q 30 )
³P
Q´ ³
1´
+
= P−
+ 2.
Q P
P
Theorem 3.11. If P :=
(3.28)
Proof. Using the equation (2.1) in the equation (2.10), we deduce
2
a P 2 b2 Q2 + a4 P 4 b4 Q4 − a6 P 6 − 48a4 P 4 b2 Q2 − 12a5 P 5 bQ
− 48a2 P 2 b4 Q4 − 76a3 P 3 b3 Q3 − b6 Q6 − 12b5 Q5 aP = 0,
(3.29)
ϕ(−q 3 )ϕ(−q 5 )
and b := a(q 2 ).
ϕ(−q)ϕ(−q 15 )
Changing q to −q in the equation (3.22), we find
where a := a(q) :=
a :=
1+P
1+Q
and b :=
.
P −1
Q−1
(3.30)
Using the equations (3.30) in the equation (3.29), we find
¡
¢¡
¢¡
− P + P Q − 1 − Q − P 2 + P 2 Q + 2P Q − Q − Q2 P 2 + P − 2P Q
¢¡
+ Q2 − P Q2 − P 4 − P 3 − Q3 − P Q2 − P Q3 + P Q4 − P 2 Q + P 2 Q3
¢¡
− P 3 Q + P 3 Q2 + P 3 Q3 + P 4 Q − Q4 + P Q − 3P 8 − P 6 − 3P 7 − P 9
− P 2 Q2 + P 2 Q3 − 5P 2 Q4 + P 3 Q2 − P 3 Q3 + 5P 3 Q4 − 5P 4 Q2 + 5P 4 Q3
+ 12P 4 Q4 + 3QP 6 + 9QP 7 + 9QP 8 + 7Q3 P 5 − 5Q3 P 6 − 3Q3 P 7 + 3Q3 P 8
292
M.S. Mahadeva Naika, S. Chandankumar, M. Harish
− 7P 5 Q2 + 3P 6 Q2 − 3P 7 Q2 − 9P 8 Q2 − 3P 9 Q2 + P 9 Q3 + 2P 5 Q4 − 7P 6 Q4
− 7P 7 Q4 + 3P 9 Q + 2P 4 Q5 − 7P 4 Q6 − 7P 4 Q7 + 5P 6 Q5 + P 6 Q6 + P 6 Q7
− 12P 5 Q5 + 5P 5 Q6 + 5P 5 Q7 + 5P 7 Q5 + P 7 Q6 + P 7 Q7 + 3P Q6 + 9P Q7
+ 9P Q8 + 7P 3 Q5 − 5P 3 Q6 − 3P 3 Q7 + 3P 3 Q8 + 3P Q9 − 7P 2 Q5 + 3P 2 Q6
¢
− 3P 2 Q7 − 9P 2 Q8 − 3P 2 Q9 + P 3 Q9 − 3Q8 − Q6 − 3Q7 − Q9 = 0.
(3.31)
As q → 0, the second factor of the equation (3.31) vanishes whereas the other
factors do not vanish. Hence, we arrive at the equation (3.28) for q ∈ (0, 1). By
analytic continuation the equation (3.28) is true for |q| < 1.
ϕ(−q 6 )ϕ(−q 10 )
ϕ(−q 3 )ϕ(−q 5 )
and Q :=
, then
15
ϕ(−q)ϕ(−q )
ϕ(−q 2 )ϕ(−q 30 )
³P
Q´ ³
1´
+
= Q−
+ 2.
(3.32)
Q P
Q
Theorem 3.12. If P :=
Proof. The proof of the equation (3.32) is similar to the proof of the equation
(3.28). Hence, we omit the details.
ϕ(−q 3 )ϕ(−q 5 )
ϕ(−q 12 )ϕ(−q 20 )
and Q :=
, then
15
ϕ(−q)ϕ(−q )
ϕ(−q 4 )ϕ(−q 60 )
³P2
³
³
³P
Q2 ´
1 ´
1 ´
Q2 ´
2
+
+
3
Q
+
−
4
P
Q
+
+
3
−
Q2
P2
Q2
PQ
Q2
P
³
´
³
´
³
1
1
1´
− P Q2 +
−
2
P
−
+
4
Q
−
= 0. (3.33)
P Q2
P
Q
Theorem 3.13. If P :=
Proof. Using the equations (2.1) and (2.2) in the equation (2.10), we deduce
48Q2 b2 P 4 a4 − Q4 b4 P 4 a4 − Q2 b2 P 2 a2 + 48Q4 b4 P 2 a2 + P 6 a6
+ 76Q3 b3 P 3 a3 + 12QbP 5 a5 + Q6 b6 + 12Q5 b5 P a = 0,
(3.34)
ψ(q 3 )ψ(q 5 )
and b := a(q 4 ).
ψ(q)ψ(q 15 )
Changing q to −q in the equation (3.22), we find
where a := a(q) :=
a :=
1+P
1+Q
and b :=
.
P −1
Q−1
Employing the equations (3.35) in the equation (3.34), we deduce
¡ 2 2
¢¡
Q P − 2QP 2 + P 2 + 2P + 1 − 4QP + 2Q − 2Q2 P + Q2 QP 2 − P 2
¢¡
− P − Q + Q2 P − Q2 3Q8 − 6P 4 − 15P 7 − 6P 8 + 45Q8 P 4 + 3QP 3
+ 18QP 4 + 45QP 5 + 60QP 6 + 45QP 7 + 18QP 8 − Q2 P + 8Q2 P 2
+ 10Q2 P 3 − 28Q2 P 4 − 68Q2 P 5 − 64Q2 P 6 − 34Q2 P 7 − 12Q2 P 8
(3.35)
Mixed modular equations involving Ramanujan’s theta-functions
293
+ Q3 P − 8Q3 P 2 − 12Q3 P 3 + 16Q3 P 4 + 38Q3 P 5 + 24Q3 P 6 + 4Q3 P 7
− 5Q4 P + 3Q4 P 2 − 9Q4 P 3 − 13Q4 P 4 + 33Q4 P 5 + 17Q4 P 6 − 19Q4 P 7
− 7Q4 P 8 − 7Q5 P + 19Q5 P 2 + 17Q5 P 3 − 33Q5 P 4 − 13Q5 P 5 + 9Q5 P 6
+ 3Q5 P 7 + 5Q5 P 8 + 4Q6 P 2 − 24Q6 P 3 + 38Q6 P 4 − 16Q6 P 5 − 12Q6 P 6
+ 8Q6 P 7 + Q6 P 8 + 3QP 9 − 3Q2 P 9 + Q3 P 9 + 3Q8 P 6 − 18Q8 P 5
+ 15Q9 P 4 − 6Q9 P 5 + Q9 P 6 + Q6 − P 3 − 20P 6 − 15P 5 − P 9 + 45Q8 P 2
+ 34Q7 P 2 + 15Q9 P 2 − 64Q7 P 3 + 68Q7 P 4 − 28Q7 P 5 − 10Q7 P 6 + 8Q7 P 7
¢
+ Q7 P 8 − 60Q8 P 3 − 20Q9 P 3 + 3Q7 + Q9 − 6Q9 P − 18Q8 P − 12Q7 P
¡ 4
P + Q4 + 3P 2 + P − 4QP − 4QP 2 + 2Q2 P − 2Q2 P 3 + 4Q3 P 2
¢
− 4Q3 P 3 − 3Q4 P + 3Q4 P 2 − Q4 P 3 + 3P 3 = 0.
(3.36)
As q → 0, the last factor of the equation (3.36) vanishes whereas the other factors
do not vanish. Hence, we arrive at the equation (3.33) for q ∈ (0, 1). By analytic
continuation the equation (3.33) is true for |q| < 1.
Acknowledgement. The authors are grateful to the referee for his/her valuable remarks and suggestions which considerably improved the quality of the paper.
REFERENCES
[1] B. C. Berndt, Ramanujan’s Notebooks, Part III, Springer-Verlag, New York, 1991.
[2] B. C. Berndt, Ramanujan’s Notebooks, Part IV, Springer-Verlag, New York, 1994.
[3] S. Bhargava, C. Adiga, M.S. Mahadeva Naika, A new class of modular equations in Ramanujan’s alternative theory of elliptic function of signature 4 and some new P –Q eta-function
identities, Indian J. Math. 45 (1) (2003), 23–39.
[4] S. Bhargava,C. Adiga, M.S. Mahadeva Naika, A new class of modular equations akin to Ramanujan’s P –Q eta-function identities and some evaluations therefrom, Adv. Stud. Contemp.
Math. 5 (1) (2002), 37–48.
[5] S. Ramanujan, Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay,
1957.
(received 14.12.2012; in revised form 24.06.2013; available online 10.09.2013)
M. S. Mahadeva Naika, M. Harish, Department of Mathematics, Bangalore University, Central
College Campus, Bengaluru-560 001, Karnataka, India
E-mail: msmnaika@rediffmail.com, numharish@gmail.com
S. Chandankumar, Department of Mathematics, Acharya Institute of Graduate Studies, Soldevanahalli, Hesarghatta Main Road, Bengaluru-560 107, Karnataka, India
E-mail: chandan.s17@gmail.com