5.0. Introduction 5.1. BasicIdentityof...

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


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 ).


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.


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


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,


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:


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)


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)


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].


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.


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).


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)


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)

.


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)


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,


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


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),


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,


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.


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.

References

Agarwal, R.P. (1999).Resonance of Ramanujan’s Mathematics, Vols. I-III, NewAge Publishers, New Delhi [Bhargava, S., Chapter 5, A look atRamanujan’swork in elliptic function theory and further development].

Berndt, B.C. (1985,89,91,94,97).Ramanujan’s Notebooks, Parts I-V, Springer,N.Y.

Berndt, B.C., Bhargava, S. and Garvan, F.G. (1995). Ramanujan’s theories ofelliptic functions to alternative bases,Trans. Amer. Math. Soc., 347, 4163-4244.

Bhargava, S. (1995). Unification of the cubic analogues of the Jacobian ellipticfunctions,J. Math. Anal. Appl., 193, 543-558.

Borwein, P.B. and Borwein, J.M. (1991). Cubic Counterpart of Jacobi’s identityand the AGM,Trans. Amer. Math. Soc., 323, 691-701.


Borwein, P.B., Borwein, J.M. and F.G. Garvan, (1994). Some cubic identities ofRamanujan,Trans. Amer. Math.Soc., 343, 35-37.

Cooper, S. (2001). The development of elliptic functions according to Ramanu-jan and Venkatachaliengar,Proc, Int. Conf. on Works of Srinivas Ramanujan(Editors: Chandrashekhar Adiga and D.D. Somasekhara) University of Mysore,Mysore 570006.

Cooper, S. Cubic elliptic functions. (Preprint).

Hardy,G.H., Ramanujan, and Chelsea, N.Y. (1978).

Hirschhorn, M.D, Garvan, F.G. and Borwein, J.M. (1993). Cubic anlogues ofthe Jacobian theta functions,Canadian J.Math., 45, 673-694.

Ramanujan, S. (1957).Notebooks I,II and III(2 volumes), TIFR, Bombay.

Ramanujan, S.(1962).Collected Works, Chalsea, N.Y.

Venkakatachaliengar, K. (1988). Development of Elliptic functions according toRamnujan, Department of Mathematics, Madurai-Kamaraj University, Madurai,Technical Report 2

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