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CHAPTER 5
RAMANUJAN’S THEORIES OF THETA AND ELLIPTICFUNCTIONS - III
[This Chapter is based on the lectures of Professor S. Bhargava of the Department of
Mathematics, University of Mysore, Manasa Gangothri, Mysore 570 006, India]
5.0. Introduction
The present lectures are aimed at covering some introductory aspects of theJacobian and Weiestrassian elliptic functions and the cubic elliptic functions im-plied in Ramanujan’s works. The lectures are a sequel to earlier lectures deliv-ered by the author in June - July 2000 and March - April 2005 SERC Schools,vide Publications 31 and 32 of Centre for Mathematical Sciences, Trivandrumand Pala Campuses respectively. It is hoped that the lectures will lead the audi-ence/ readers to further reading and research.
5.1. Basic Identity of Ramanujan and WeierstrassianTheory of Elliptic Functions
Following is an identity of Ramanujan which plays an important role in hisdevelopment of elliptic function theory [9, 11, 13].
Theorem 5.1.1. If q = eiτ and0 < Im θ < Im τ, then
[14
cotθ
2+
∞∑
1
qn sinnθ1− qn
]2
=
(14
cotθ
2
)2
+
(12
∞∑
1
nqn
1− qn
)
221
222 5. RAMANUJAN’S THEORIES OF THETA AND ELLIPTIC FUNCTIONS- III
+
∞∑
1
{
qn
(1− qn)2− nqn
2(1− qn)
}
cosnθ. (5.1.1)
or, with ξ = eiθ,
( ∞∑
−∞
′ ξn
1− qn
)2
= −2∞∑
1
nqn
1− qn+
∞∑
1
(n+ 1)(ξn + qnξ−n)(1− qn)
−∞∑
1
(ξn + qnξ−n)(1− qn)2
.
(5.1.2)
Proof 5.1.1. The following proof of Ramanujan is elementary. [12, p.138].
On using the elementary identity
cotθ
2sinnθ = (1+ cosnθ) + 2
n−1∑
m=1
cosmθ
we have
[14
cotθ
2+
∞∑
1
qn sinnθ1− qn
]2
=
(14
cotθ
2
)2
+12
∞∑
1
qn sinnθ cot θ21− qn
+
∞∑
1
∞∑
1
qm+n sinmθ sinnθ(1− qm)(1− qn)
=
(14
cotθ
2
)2
+
∞∑
0
cncosnθ
where
C0 =12+
∞∑
1
qn
1− qn+
12
∞∑
1
( qn
1− qn
)2
=12
∞∑
1
nqn
1− qn
5.1. BASIC IDENTITY OF RAMANUJAN AND WEIERSTRASSIAN THEORY 223
as required in (5.1.1). Further forn ≥ 1,
cn=12
qn
1− qn+
∞∑
1
qn+r
1− qn+r+
∞∑
1
( qr
1− qr
)( qn+r
1− qn+r
)
− 12
n−1∑
1
qr
1− qr
qn−r
1− qn−r
which reduces to the required expression in (5.1.1) on some manipulations. Weomit the details.
Exercises 5.1.
5.1.1. Show in the proof of Theorem 5.1.1 thatCn indeed equals
{
qn
(1− qn)2− nqn
2(1− qn)
}
, n ≥ 1.
Remark 5.1.1 (9, p135.). Identity (5.1.1) is indeed equivalent to the followingidentity in the Weierstrassian elliptic function theory:
{
ζ(θ) − η1θ
π
}2
− p(θ) = −16+ 4
∞∑
1
qm cosmθ(1− qm)2
(5.1.3)
where
ζ(θ) :=12
cotθ
2+ 2
∞∑
1
qn sinnθ1− qn
+ θ
112− 2
∞∑
1
qn
(1− qn)2
or, with z= eiθ
224 5. RAMANUJAN’S THEORIES OF THETA AND ELLIPTIC FUNCTIONS- III
ζ(θ) :=12i
1+ z1− z
+ 2∞∑
1
zqn
1− zqn− 2
∞∑
1
z−1qn
1− z−1qn
+ θ
112− 2
∞∑
1
qn
(1− qn)2
(5.1.4)
:= −iρ1(z) +(
θ
12
)
P(q) (5.1.5)
and
p(θ) := −ζ′(θ) = 14
cosec2θ
2− 2
∞∑
1
nqn cosnθ1− qn
+ 2∞∑
1
qn
(1− qn)2− 1
12
= −zρ′1(z) −P12
(5.1.6)
or
p(θ) := −∞∑
−∞
zqn
(1− zqn)2+ 2
∞∑
1
qn
(1− qn)2− 1
12, z= eiθ (5.1.7)
and
η1 := π
112− 2
∞∑
1
qn
(1− qn)2
= ζ(π).
Definition 5.1.1. The functionsp(θ) andζ(θ) are are respectively the Weier-strassian elliptic function and the Weierstrassian zeta function.
Exercises 5.1.
5.1.2. Prove double periodicity ofp(θ) and identify the singularities.5.1.3. Settingη2 = ζ(πτ), show thatη2 = τη1 − i
2 (Legendre’s formula ).
5.1. BASIC IDENTITY OF RAMANUJAN AND WEIERSTRASSIAN THEORY 225
Theorem 5.1.2. We can rewrite p(θ) as
p(θ) =1θ2+
′∑
m,n
[ 1(θ − 2πn− 2πτm)2
− 1(2πn+ 2πτm)2
]
(5.1.8)
with q= e2πiτ, which is the classical form of Weierstrassian elliptic function.
Proof 5.1.2. We have from Definition 5.1.1 that
p(θ) = − 112+ 2
∞∑
1
1(eπimτ − e−πimτ)2
−∞∑
−∞
1
(ei θ2+iπτn − e−iθ2−iπτn)2
= − 112− 1
2
∞∑
1
1
sin2 πmτ+
14
∞∑
−∞
1
sin2( θ2 + πτm)
=1
4 sin2 θ2
− 112− 1
4
∞∑
−∞
′ 1
sin2 mπτ+
14
∞∑
−∞
1
sin2( θ2 +mπτ)
=1
4 sin2 θ2
− 112+
14
∞∑
−∞
′( 1
sin2( θ2 +mπτ)− 1
sin2 mπτ
)
=14
∞∑
−∞
1
( θ2 − πn)2− 1
123π2
∞∑
−∞
′ 1n2
+14
∞∑
m=−∞
′∞∑
n=−∞
[ 1
( θ2 +mπτ − nπ)2− 1
(mπτ − nπ)2
]
=1θ2+
∞∑
−∞
′[ 1(θ − 2πn)2
− 1(2πn)2
]
+
∞∑
m=−∞
′∞∑
n=−∞
[ 1(θ − 2mπτ − 2nπ)2
− 1(2mπτ + 2nπ)2
]
=1θ2+
∞∑
−∞
∑
(m,n),(0,0)
( 1(θ − 2mπτ − 2nπ)2
− 1(2mπτ + 2nπ)2
)
.
This proves the theorem.
226 5. RAMANUJAN’S THEORIES OF THETA AND ELLIPTIC FUNCTIONS- III
Theorem 5.1.3. [13] (Generalization of basic identity of Ramanujan)Let |q| < |z| < 1 and let
ρ1(z) :=12+
∞∑
−∞
′ zn
1− qn, (5.1.9)
as before, and let
ρ2(z) := − 112+
∞∑
−∞
′ qnzn
(1− qn)2. (5.1.10)
Then
ρ1(α)ρ1(β) + ρ1(β)ρ1(γ) + ρ1(γ)ρ1(α) = ρ2(α) + ρ2(β) + ρ2(γ) (5.1.11)
for all complexα, β andγ with αβγ = 1.
Proof 5.1.3. We can rewriteρ1(z) andρ2(z) in their global forms on slight ma-nipulations:
ρ1(z) =12
(1+ z1− z
)
+
∞∑
1
qn(zn − z−n)1− qn
(5.1.12)
=12
(1+ z1− z
)
+
∞∑
1
qnz1− qnz
−∞∑
1
qnz−1
1− qnz−1(5.1.13)
and
ρ2(z) = −112+
∞∑
1
qn(zn + z−n)(1− qn)2
(5.1.14)
= − 112+
∞∑
1
nqnz1− qnz
+
∞∑
1
nqnz−1
1− qnz−1. (5.1.15)
With these global forms (5.1.11) would be valid globally exceptαβγ = 1. Weeasily see from (5.1.12) and (5.1.14) that
5.1. BASIC IDENTITY OF RAMANUJAN AND WEIERSTRASSIAN THEORY 227
ρ1
(
1z
)
= −ρ1(z) and ρ2
(
1z
)
= ρ2(z). (5.1.16)
Employing (5.1.16) we can expand each side of (5.1.11) into power series inαandβ and then see that the two sides are equal. We omit details.
Corollary 5.1.1. Identity (5.1.11) can be rewritten as
ρ2(α) + ρ2(β) + ρ2(αβ) = (1− β) ρ1(β)
{
ρ1(αβ) − ρ1(α)β − 1
}
− ρ1(αβ) ρ1(α).
Lettingβ→ 1 we get
2ρ2(α) + ρ2(1) = αρ′1(α) − ρ21(α). (5.1.17)
This is indeed the same as Ramanujan’s basic identity (5.1.1).
Exercises 5.1.
5.1.4. Complete the details in the proof of Theorem 5.1.3.
5.1.5. Prove the equivalence of equations (5.1.17) and (5.1.1).
Theorem 5.1.4. (Ramanujan’s basic identity and addition theorem for theWeierstrassian elliptic function). The following holds
p(a+ b) = −p(a) − p(b) +14
[ p′(a) − p′(b)p(a) − p(b)
]2
. (5.1.18)
Proof 5.1.4. From (5.1.17) (or what is the same (5.1.1)) and on using (5.1.11)we have, forαβγ = 1,
(ρ1(α) + ρ1(β) + ρ1(γ))2 =∑
αρ′1(α) − 3ρ2(1) (5.1.19’)
and hence, on differentiating with respect toβ,
228 5. RAMANUJAN’S THEORIES OF THETA AND ELLIPTIC FUNCTIONS- III
2(∑
ρ1(α))
(
ρ′1(β) − γβρ′1(γ)
)
= βρ′′1 (β) +[
γρ′′1 (γ) + ρ′1(γ)]
(
−γβ
)
+ ρ′1(β)
or,
2(∑
ρ1(α))
(
βρ′1(β) − γρ′1(γ))
= β2ρ′′1 (β) − γ2ρ′′1 (γ) − γρ′1(γ) + βρ′1(β)(5.1.19)
or,
∑
ρ1(α) =12
[
β2ρ′′1 (β) − γ2ρ′′1 (γ)
βρ′1(β) − γρ′1(γ)+ 1
]
. (5.1.20)
Now, from (5.1.6),
p′(a) = −i[α2ρ′′1 (α) + αρ′1(α)], α = eia
p′(b) = −i[β2ρ′′1 (β) + βρ′1(β)], β = eib
p′(c) = −i[γ2ρ′′1 (γ) + γρ′1(γ)], γ = eic, αβγ = 1,
so that,
p′(b) − p′(c) = −i[β2ρ′′1 (β) − γ2ρ′′1 (γ) + βρ′1(β) − γρ′1(γ)].
Using (5.1.6) again, the last equation gives
p′(b) − p′(c)p(b) − p(c)
=
[
β2ρ′′1 (β) − γ2ρ′′1 (γ)
βρ′1(β) − γρ′1(γ)+ 1
]
.
From (5.1.19’) and (5.1.20), we have
5.2. BASIC IDENTITY OF RAMANUJAN AND JACOBI’S ELLIPTIC FUNCTIONS 229
14
(
p′(b) − p′(c)p(b) − p(c)
)2
=(∑
ρ1(α))2
= 3ρ2(1)−∑
αρ′1(α)
= 3ρ2(1)− αρ′1(α) − βρ′1(β) − γρ′1(γ)
= p(a) + p(b) + p(c), on using (5.1.6) and (5.1.10),
= p(−b− c) + p(b) + p(c)
= p(b+ c) + p(b) + p(c), since p(θ) is even by (5.1.8).
This is the same as (5.1.18) but for the arguments.
5.2. Basic Identity of Ramanujan and Jacobi’s Ellip-tic Functions
Definition 5.2.1. Ramanujan [2][11, Second Notebook, Chapter 18] defines
S :=∞∑
0
sin(2n+ 1)θ2sinh(2n+ 1)y2
C :=∞∑
0
cos(2n+ 1)θ2cosh(2n+ 1)y2
and
C1 :=12+
∞∑
1
cosnθcoshny
and are infact the Jacobian elliptic functions (in their Fourier series form)sn, cnanddn:
230 5. RAMANUJAN’S THEORIES OF THETA AND ELLIPTIC FUNCTIONS- III
sn
(
12
zθ
)
:=2
z√
xS :=
2
z√
x
∞∑
0
sin(2n+ 1)θ2sinh(2n+ 1)y2
cn
(
12
zθ
)
:=2
z√
xC :=
2
z√
x
∞∑
0
cos(2n+ 1)θ2cosh(2n+ 1)y2
and
dn
(
12
zθ
)
:=2z
C1 :=2z
12+
∞∑
1
cosnθcoshny
or, in standard notations and complex form,
sn(Kπθ, k
)
:= − iπKk
eπiτ2 +
iθ2
∞∑
−∞
(qζ)n
1− q2n+1
= − iπKk
eπiτ2 +
iθ2 f (q, qζ)
= − iπKk
eπiτ2 +
iθ2 f (eπiτ, eπiτ+iθ)
cn(Kθπ, k
)
:=π
Kkeπiτ2 +
iθ2 f (−q, qζ)
=π
Kkeπiτ2 +
iθ2
∞∑
−∞
(qζ)n
1+ q2n+1
=π
Kkeπiτ2 +
iθ2 f (−eπiτ, qπiτ+iθ)
and
dn(Kθπ, k
)
=2z
C1 =π
Kf (−1, qζ) =
π
K
∞∑
−∞
(qζ)η
1+ q2n=π
Kf (eiπ, eiπτ+iθ)
5.2. BASIC IDENTITY OF RAMANUJAN AND JACOBI’S ELLIPTIC FUNCTIONS 231
where
f (a, t) =∞∑
−∞
tn
1− aq2n, |q| < 1, a , q2n, n = 0,±1, · · ·
q = eiπτ = e−y,
ζ = eiθ
x = k2 := 16q(−q2; q2)8∞/(−q; q2)8
∞
2Kπ
:= z :=
∞∑
−∞qn2
= (−q; q2)4∞(q2; q2)2
∞.
Here, as usual,
(a; q)∞ := Π∞0 (1− aqn), |q| < 1
(a; q)n := (a; q)∞/(aqn; q)∞
=
1 if n = 0
(1− a)(1− aq) · · · (1− aqn−1) if n is a positive integer.
Theorem 5.2.1. (Ramanujan) [11, Second Notebook, Chapter 18].
C2 + S2 =xz2
4or cn2 + sn2 = 1
C21 + S2 =
z2
4or dn2 + k2sn2 = 1
and
sncn=π2
K2k2CS =
π2
K2k2
∞∑
1
nsinnθcoshny
.
CS+dC1
dθ= 0 = C1S +
dCdθ= CC1 −
dSdθ.
Further, if for 0 ≤ φ < 2π(
z√
x2
cn=
)
C =z√
x2
cosφ (i.e., cn(Kπθ, k
)
= cosφ)
232 5. RAMANUJAN’S THEORIES OF THETA AND ELLIPTIC FUNCTIONS- III
and(
z√
x2
sn=
)
S =z√
x2
sinφ (i.e., sn(Kπθ, k
)
= sinφ).
then( z2
dn=)
C1 =z2
√
1− xsin2 φ
that is
dn=√
1− k2sn2
z2
cosφ√
1− xsin2 φ =ddθ
sinφ = cosφdφdθ
that is
Kπ
cn√
1− k2sn2 =ddθ
sn(Kπθ, k
)
= cn(Kπθ, k
) dφdθ
and
θ =2z
∫ φ
0
dφ√
1− xsin2φ
that is
Kπ
dθdφ=
1√
1− k2sn2=
1dn
ordφdθ=
Kπ
dn(Kθπ, k
)
.
Proof 5.2.1. We omit the proof but direct the readers to [13] and to the author’scontributory Chapter to R.P. Agarwal’s book [1]. We may justmention here thatRamanujan’s basic identity (5.1.1) directly yields the first two identities whilethe others follow easily.
Exercises 5.2.
5.2.1. Learn the proof of Theorem 5.2.1 from [1. Chapter 5 by S. Bhargava].
5.2. BASIC IDENTITY OF RAMANUJAN AND JACOBI’S ELLIPTIC FUNCTIONS 233
Theorem 5.2.2. The following product forms hold for the Jacobian ellipticfunctions C,S and C1.
S =Kkπ
sn(Kθπ, k
)
=−i(qζ)
12 (q2ζ; q2)∞(ζ−1; q2)∞(q2; q2)2
∞(qζ; q2)∞(qζ−1; q2)∞(q; q2)2
∞
C =Kkπ
cn(Kθπ, k
)
=(qζ)
12 (−q2ζ; q2)∞(−ζ−1; q2)∞(q2; q2)2
∞(qζ; q2)∞(qζ−1; q2)∞(−q; q2)2
∞
and
C1 =Kπ
dn(Kθπ, k
)
=(−qζ; q2)∞(−qζ−1; q2)∞(q2; q2)2
∞2(qζ; q2)∞(qζ−1; q2)∞(−q2; q2)2
∞.
Proof 5.2.2. We recall the “remarkable”1ψ1− summation formula of Ramanu-jan [9,11,13]
∞∑
−∞
(α−1; q2)n(−αqz)n
(βq2; q2)n=
∞∑
−∞
(β−1; q2)n(−βqz )n
(αq2; q2)n
=(−qz; q2)∞(−q
z; q2)∞(q2; q2)∞(αβq2; q2)∞
(−αqz; q2)∞(−βqz ; q2)∞(αq2; q2)∞(βq2; q2)∞
.
Puttingα−1 = β and changingz to ζ in this we get
∞∑
−∞
(−qζβ
)n
1− βq2n=
(−qζ; q2)∞(−qζ−1; q2)∞(q2; q2)2∞
(−qζβ
; q2)∞(−βqζ
; q2)∞(βq2; q2)∞(q2
β; q2)∞
.
Puttingβ = q, changingζ to −ζq in this and multiplying throughout by−i(ζq)12
we get the first of the required identities. Similarlyβ → −q, ζ → ζq gives thesecond andβ→ −1 gives the third.
234 5. RAMANUJAN’S THEORIES OF THETA AND ELLIPTIC FUNCTIONS- III
Exercises 5.2.
5.2.2. Work out the details in the proof of Theorem 5.2.2.
5.2.3. With f (a, t) as before, put (following S. Cooper [7])
f1(θ) := −i f (eiπ, eiθ)
f2(θ) := −iei θ2 f (eiπτ, eiθ)
and
f3(θ) := −iei θ2 f (eiπ+iπτ, eiθ).
Then show that they are respectively the Jacobi’s elliptic functionsKπcs
(
Kθπ, k
)
, Kπns
(
Kθπ, k
)
and Kπds
(
Kθπ, k
)
. In other words, show that
f1(θ) =12
cotθ
2(q2; q2)2
∞(−q2; q2)2
∞
∞∏
n=1
(1+ 2q2n cosθ + q4n)(1− 2q2n cosθ + q4n)
=12
cotθ
2− 2
∞∑
1
q2n
1+ q2nsinnθ
f2(θ) =12
cosecθ
2(q2; q2)2
∞(−q2; q2)2
∞
∞∏
n=1
(1− 2q2n−1 cosθ + q4n−2)(1− 2q2n cosθ + q4n)
=12
cosecθ
2+ 2
∞∑
n=0
q2n+1
1− q2n+1sin(n+
12
)θ
f3(θ) =12
cosecθ
2(q2; q2)2
∞(−q2; q2)2
∞
∞∏
n=1
(1+ 2q2n−1 cosθ + q4n−2)(1− 2q2n cosθ + q4n)
=12
cosecθ
2− 2
∞∑
n=0
q2n+1
1+ q2n+1sin(n+
12
)θ.
5.2.4. Write out the product forms forS,C andC1 obtained in Theorem 5.2.2in respective trigonometric forms as in Exercise 5.2.3
5.3. VENKATACHALIENGAR’S GENERALIZATION OF RAMANUJAN’S FUNDAMENTAL ...235
5.3. Venkatachaliengar’s Generalization of Ramanu-jan’s Fundamental Identity and Relations Be-tween Jacobian’s and Weierstrassian Elliptic Func-tions
The following theorem is a generalization due to K. Venkatachaliengar [13]of Ramanujan’s basic identity (5.1.1) and is crucial to further development ofRamunajan’s theory of elliptic functions.
Theorem 5.3.1. (Fundamental multiplicative identity)If f (a, t) is as in Definition 5.2.1, we have
f (x, y) f (x, z) = x∂
∂xf (x, yz) + f (x, yz)(ρ1(y) + ρ2(z)) (5.3.1)
ρ1 being as in (5.1.9).
Proof 5.3.1. We have
f (x, y) f (x, z) =∞∑
−∞
∞∑
−∞
ynzm
(1− xq2n)(1− xq2m)
=
∞∑
−∞
(yz)n
(1− xq2n)2+
∞∑
−∞m,n
∞∑
−∞
ynzm
(1− xq2m)(1− xq2n).
It is enough to show that the first sum and the second sum in the last identityequal respectively the first and second terms of the right side of the identity to beproved except perhaps for mutually canceling terms. This wedo now. Firstly,
∞∑
−∞
(yz)n
(1− xq2m)2=
∞∑
−∞
∂
∂x
(
yzq2
)m
(1− xq2m)=
∂
∂x
∞∑
−∞
(
yzq2
)m
(1− xq2m)
=∂
∂xf
(
yzq2, x
)
=∂
∂x[x f(y, zx)] = x
∂
∂xf (yz, x) + f (yz, x)
on employing the easily proved identity
f (yzq−2, x) = x f(yz, x).
236 5. RAMANUJAN’S THEORIES OF THETA AND ELLIPTIC FUNCTIONS- III
Finally,
∑
m,n
∑ ynzm
(1− xq2n)(1− xq2m)=
∞∑
m=−∞
∑
k
′ ym+k zm
(1− xq2m+2k)(1− xq2m)
=
∞∑
−∞
∑
k
′ ym+k zm
(1− q2k)(1− xq2m+2k)+
∞∑
−∞
∑
k
′ ym+k zm
(1− q−2k)(1− xq2m)
=
∞∑
−∞
∑
k
′ (yz)m+k
(1− xq2(m+k))z−k
(1− q2k)+
∞∑
−∞
∑
k
′ (yz)m
(1− xq2m)yk
(1− q−2k)
=
(
−12+ ρ1(z
−1)
)
f (yz, x) +
(
−12+ ρ1(y
−1)
)
f (yz, x)
= (ρ1(y) + ρ1(z)) f (yz, x) − f (yz, x)
on changingm+ k to m in the first sum andk to −k in the second sum and usingthe trivial property
ρ1(z) = ρ1(z−1) of ρ1.
Corollary 5.3.1. [7]
f (eiα, eiθ) f (e−iα, eiθ) = p(α) − p(θ). (5.3.2)
Proof 5.3.2. Letting y→ 1z in (5.3.1), we have
f (x, y) f (x, y−1) = xddx
ρ1(x) − y ddy
ρ1(y). (5.3.3)
For,
limy→ 1
z
x∂
∂x
∞∑
−∞
xn
1− yzq2n=
∞∑
−∞
′ nxn
1− q2n= x
ddx
ρ1(x)
5.3. VENKATACHALIENGAR’S GENERALIZATION OF RAMANUJAN’S FUNDAMENTAL ...237
and
limy→ 1
z
f (x, yz) (ρ1(y) + ρ1(z)) = limy→ 1
z
(1− yz) f (yz, x) limy→ 1
z
[
ρ1(y) + ρ1(z)1− yz
]
= (−1) limy→ 1
z
ρ1(y) − ρ1(1z)
y − 1z
1z
= −y ρ′1(y).
Now (5.3.3) implies (5.3.2) on puttingx = eiθ andy = eiα on using (5.1.6).
Corollary 5.3.2. [7]
(
f 21 (θ) =
) K2
π2cs2
(Kθπ, k
)
= p(θ) − e1 (5.3.4)
(
f 22 (θ) =
) K2
π2ns2
(Kθπ, k
)
= p(θ) − e2 (5.3.5)
(
f 23 (θ) =
) K2
π2ds2
(Kθπ, k
)
= p(θ) − e3 (5.3.6)
where
e1 = p(π), e2 = p(πτ) and e3 = p(π + πτ).
Further,
e1 − e2 =14
(−q; q2)4∞ (q2; q2)4
∞(q; q2)4
∞ (−q2; q2)4∞= f 2
2 (θ) − f 21 (θ) =
K2
π2(ns2 − cs2) (5.3.7)
e3 − e2 = 4q(−q2; q2)4
∞(q2; q2)4∞
(−q; q2)4∞ (q; q2)4
∞= f 2
2 (θ) − f 23 (θ) =
K2
π2((ns)2 − (ds)2)
(5.3.8)
and
e1 − e3 =14
(q; q2)4∞(q2; q2)4
∞(−q2; q4)4
∞ (−q; q2)4∞= f 2
3 (θ) − f 21 (θ) =
K2
π2(cs2 − ds2) (5.3.9)
238 5. RAMANUJAN’S THEORIES OF THETA AND ELLIPTIC FUNCTIONS- III
Proof 5.3.3. Puttingα = π in (5.3.2) and using the results of Exercise 5.2.3 gives(5.3.4). Similarly (5.3.5) and (5.3.6) follow. Puttingθ = π in (5.3.5),θ = πτ + πin (5.3.5) andθ = π in (5.3.6) and using the results of Exercise 5.2.3 again givesfirst half of (5.3.7) - (5.3.9) respectively. For the remaining identities, simply use(5.3.4) - (5.3.6).
Corollary 5.3.3. [7] From (5.3.1) we have,
f (−1, eiθ) f (q, eiθ) =1i∂
∂θf (−q, eiθ) +
12
f (−q, eiθ)
since, from the definition ofρ1(z) we haveρ1(eiπ) = 0 andρ1(eiπτ) = 12. Thus, on
using the results of Exercise 5.2.3 and simplifying,
−eiθ2 f1(θ) f2(θ) = e
iθ2 f ′3(θ)
or,
f ′3(θ) = − f1(θ) f2(θ).
Similarly,
f ′1(θ) = − f2(θ) f3(θ)
and
f ′2(θ) = − f3(θ) f1(θ).
Employing results of Corollary 5.3.2, these reduce to
[p′(θ)]2 =
3∏
j=1
(p(θ) − ej).
Exercises 5.3.
5.3.1. [7] [Addition Theorems for the Jacobian elliptic functions]Employing (5.3.1) show that
5.4. CUBIC ELLIPTIC FUNCTION 239
∂
∂αf (eiα, eiθ) f (eiβ, eiθ) − ∂
∂βf (eiα, eiθ) f (eiβ, eiθ)
= f (ei(α+β), eiθ)
(
ddαρ1(e
iα) − ddβρ1(e
iβ)
)
and hence show (on employing the definitions off1, f2, f3 andp):
f1(α + β) =f1(α) f2(β) f3(β) − f1(β) f2(α) f3(α)
f 21 (β) − f 2
1 (α)
and two similar formulas.
5.4. Cubic Elliptic Function
The following theorem of Ramanujan [13, Second Notebook p.257] providesa cubic analogue of Jacobian elliptic functions, namely thefunction given by(5.4.3).
Theorem 5.4.1. [13, p257]Let, for 0 < x < 1,
z := 2F1
(
13,23
; 1; x
)
,
2F1(a, b; c; x) := 1+∞∑
1
[a]n[b]n
[c]nn!, [a]n = a(a+ 1) · · · (a+ n− 1), (a)0 = 1, a , 0
q := e−y,
y :=2π√
3
2F1(13,
23; 1; 1− x)
2F1(13,
23; 1; x)
.
240 5. RAMANUJAN’S THEORIES OF THETA AND ELLIPTIC FUNCTIONS- III
For 0 ≤ φ ≤ π2, define θ = θ(φ) by
θz=∫ φ
02F1
(
13,23
;12
; xsin2 t
)
dt, (5.4.1)
or, equivalently , by
θz=∫ φ
0
cos(
13 sin−1
(
√
xsin2 t))
dt√
1− xsin2 t. (5.4.2)
Then, for0 ≤ θ ≤ π2, the following inversion holds
φ = θ + 3∞∑
n=1
sin(2nθ)n(1+ 2 cosh(ny))
= θ + 3∞∑
n=1
sin(2nθ)qn
n(1+ q+ qn). (5.4.3)
The integral and the inverse are clearly analogous to the classical elliptic inte-gral and one of classical Jacobi’s elliptic functions.
Proof 5.4.1. Since the proof is protracted, we will be brief. For further detailsone may see the references [3, 8]. In what follows (small case) z stands forcomplex number unlike before and thez used hitherto is replaced by (Cap)Z.Define
v(z, q) := 1+ 3∞∑
n=1
(zn + z−n)qn
1+ qn + q2n, |q| < |z| < |q|−1, (5.4.4)
or, globally
v(z, q) := 1+ 3∞∑
n=0
{
zq3n+1
1− zq3n+1− zq3n+2
1− zq3n+2+
z−1q3n+1
1− z−1q3n+1− z−1q3n+2
1− z−1q3n+2
}
(5.4.5)
5.4. CUBIC ELLIPTIC FUNCTION 241
and
V(θ) := v(eziθ, q) = 1+ 6∞∑
n=1
cos(2nθ)qn
1+ qn + q2n. (5.4.6)
• Firstly, we can have the following representations (i)-(ii).
(i) v in terms of the cubic theta functions:
v (eiθ, q) =32
(q; q)2∞(q3; q3)2
∞(q2; q2)∞(q6; q6)∞
b(q,−eiθ)b(q, eiθ)
− 12
b(q)2
b(q2)(5.4.7)
whereb(q) is the one-variable cubic theta function [4,5,6,10] givenby
b(q) :=∞∑
−∞
∞∑
−∞qm2+mn+n2
wm−n =(q; q)3
∞(q3; q3)∞
(5.4.8)
andb(q, z) is the two-variable cubic theta function [4,5,6,10] givenby
b(q, z) := (q; q)∞(q3; q3)∞(qz; q)∞(qz−1; q)∞
(q3z; q3)∞(q3z−1; q3)∞, (5.4.9)
the other associated cubic theta functions being,
242 5. RAMANUJAN’S THEORIES OF THETA AND ELLIPTIC FUNCTIONS- III
a(q) :=∞∑
−∞
∞∑
−∞qm2+mn+n2
= 1+ 6∞∑
n=1
(
q3n−2
1− q3n−2− q3n−1
1− q3n−1
)
= 1+ 6∞∑
n=1
qn
1+ qn + q2n, (5.4.10)
c(q) :=∞∑
−∞
∞∑
−∞q(m+ 1
3 )2+(m+ 13 )(n+ 1
3 )+(n+ 13 )2
= 3q13(q3; q3)∞(q; q)∞
(5.4.11)
a(q, z) :=∞∑
−∞
∞∑
−∞qm2+mn+n2
zm−n (5.4.12)
b(q, z) :=∞∑
−∞
∞∑
−∞qm2+mn+n2
ωm−nzn
= (q; q)∞(q3; q3)∞(qz; q)∞(qz−1; q)∞
(q3z; q3)∞(q3z−1; q3)∞(5.4.13)
c(q, z) :=∞∑
−∞
∞∑
−∞q(m+ 1
3 )2+(m+ 13 )(n+ 1
3 )+(n+ 13 )2
zm−n
= q13 (q; q)∞(q3; q3)∞(1+ z+ z−1)
(q3z3; q3)∞(q3z−3; q3)∞(qz; q)∞(qz−1; q)∞
(5.4.14)
(ii)dVdθ=
∂
∂θv(eiθ, q) = q(z− z−1) (5.4.15)
×(z2q3; q3)∞(z−2q3; q3)∞(q; q3)∞(q2; q3)∞(q3; q3)4
∞(zq; q3)2
∞(z−1q; q3)2∞(zq2; q3)2
∞(z−1q2; q3)2∞
(5.4.16)
For proofs of (i)-(ii) it is best to study [3] and [8], the proofs given in the latterreference being simpler where the author studiesg1(θ, q) = 1
6v(eiθ, q) and associ-
atedg2(θ, q). His proof employs theory of elliptic functions directly and realizes
5.4. CUBIC ELLIPTIC FUNCTION 243
v(eiθ, q) to be doubly periodic meromorphic function and hence elliptic.
• Next we define
ψ(θ) :=14x
(
4− V3(θ)Z3− 3
V2(θ)Z2
)
=1
4xZ3(Z − V)(2Z + V)2 (5.4.17)
so that
dψ(θ)dθ= − 3V
4xZ3(V + 2Z)
dVdθ. (5.4.18)
We now wish to show
(
dψdθ
)2
= 4ψ(1− ψ)V2, (5.4.19)
or, on using (5.4.17) and (5.4.18),
81V2
16x2z2(V + 2Z)2
(
dVdθ
)2
= 4V2
4x
(
4− V3
Z3− 3
V2
Z2
) (
1− 14x
(
4− V3
Z3− 3
V2
Z2
))
or,
81
(
dVdθ
)2
= (Z − V)(4xZ3 − (Z − V)(V + 2Z)2) (5.4.19)′
or on using (5.4.15),
244 5. RAMANUJAN’S THEORIES OF THETA AND ELLIPTIC FUNCTIONS- III
81q2(z− z−1)2(z2q3; q3)2∞(z−2q3; q3)2
∞(q; q)2∞(q3; q3)6
∞
= (zq; q3)4∞(z−1q; q3)4
∞(zq2; q3)4∞(z−1q2; q3)4
∞
× (Z − V)(4xZ3 − (Z − V)(V + 2Z)2)
or, using (5.4.9),
81q2(z− z−1)2(z2q3; q3)2∞(z−2q3; q3)2
∞(q; q)6∞(q3; q3)10
∞
= b4(q; z)(Z − V)(4xZ3 − (Z − V)(V + 2Z)2). (5.4.19)′′
For the rest of the proof of (5.4.19), which is protracted, werefer to [3].
• Now it is not difficult to argue, on employing the many properties ofV(θ) proved so far, that
0 < ψ < 1, V > 0,dVdθ
< 0,dψdθ
> 0, in 0 < θ <π
2, ψ(0) = 0, ψ(
π
2) = 1.
(5.4.20)
Hence,
V(θ) =1
2√
ψ(θ)√
1− ψ(θ)
dψdθ, 0 < θ <
π
2
or
∫ θ
0V(t)dt =
12
∫ θ
0
1
2√
ψ(t)(1− ψ(t))
dψdt· dt, 0 < θ <
π
2,
5.4. CUBIC ELLIPTIC FUNCTION 245
or, with
V(θ) =:dΦ(θ)
dθ(5.4.21)
and using (5.4.6),
Φ(θ) = θ + 6∞∑
1
sin(2nθ)qn
n(1+ qn + q2n)=
12
∫ ψ(θ)
0
du√
u(1− u)= sin−1
( √
ψ(θ))
(5.4.22)or, on using (5.4.17),
4xsin2(Φ(θ)) = 4xψ(θ) = 4− V3
Z3− 3
V2
Z2, 0 ≤ θ ≤ π
2,
or, on puttingS(x) : =ZV, (5.4.23)
4(1− xsin2Φ(θ)) S3(x) − 3S(x) − 1 = 0 (5.4.24)
whereS(x) is continuous in (−1, 1) andS(0) = 1. But,
S(x) = 2F1
(
13,23
;12
; xsin2Φ(θ)
)
=(
1− xsin2Φ(θ))− 1
2 cos
(
13
sin−1(sinΦ(θ)√
x)
)
is the unique solution of the cubic (5.4.24) as can be easily verified.
• Thus, we have, on using (5.4.21) and (5.4.23).
ZV= z
dΘ(φ)dφ
= 2F1
(
13,23
;12
; xsin2 φ
)
whereΘ :[
0, π2]
→[
0, π2]
is the set theoretic inverse ofΦ. Hence,
Zθ =∫ φ
02F1
(
13,23
;12
; xsin2 φ
)
dφ.
This establishes the theorem.
246 5. RAMANUJAN’S THEORIES OF THETA AND ELLIPTIC FUNCTIONS- III
Exercises 5.4.
5.4.1. Show that the integrals on the right sides of (5.4.1) and (5.4.2) are equal.
5.4.2. Obtain (5.4.5) from (5.4.4).
5.4.3. Obtain the form (5.4.19)” of (5.4.19)’ in detail.
5.4.4. Prove the various properties ofV andψ given by (5.4.19).
Acknowledgment: The author is thankful to Dr. K.R. Vasuki, Professor andHead. Department of M.C.A. and Professor of Mathematics, Centre for Researchin Mathematics Acharya Institute of Technology, Bangalorefor hospitality en-abling write-up of this notes.
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Bhargava, S. (1995). Unification of the cubic analogues of the Jacobian ellipticfunctions,J. Math. Anal. Appl., 193, 543-558.
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5.4. CUBIC ELLIPTIC FUNCTION 247
Borwein, P.B., Borwein, J.M. and F.G. Garvan, (1994). Some cubic identities ofRamanujan,Trans. Amer. Math.Soc., 343, 35-37.
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