Ramanujan’s theories of elliptic functions to alternative...

Ramanujan’s theories of elliptic functions toalternative bases, and beyond.

Shaun Cooper

Massey University, Auckland

Askey 80 Conference. December 6, 2013.

Outline

1 Sporadic sequences

2 Background: classical theory

3 Ramanujan’s alternative theories

4 Beyond Ramanujan’s alternative theories

Roger Apery, 1916–1994

Classical results∞�

n=1

1

n2=

π2

6,

∞�

n=1

1

n4=

π4

90,

∞�

n=1

1

n6=

π6

945, · · ·

Apery’s theorem (1978)

ζ(3) :=∞�

n=1

1

n3is irrational

Apery introduced two sequences of integers: one to prove ζ(2) �∈ Q(well-known) and the other to prove ζ(3) �∈ Q (new).

Apery’s numbers used to prove ζ(2) �∈ Q

(k + 1)2sk+1 = (11k2 + 11k + 3)sk + k2sk−1, s0 = 1

The numbers sk are integers—not obvious.


(k + 1)3tk+1 = (2k + 1)(17k2 + 17k + 5)tk − k3tk−1, t0 = 1

The numbers tk are integers—not obvious.


(k + 1)2sk+1 = (11k2 + 11k + 3)sk + k2sk−1, s0 = 1

sk =k�

j=0

�k

j

�2�k + j

j

�

Proof: by the method of creative telescoping. Now, automated.

The numbers sk are integers—now obvious from the binomial sum.

F. Beukers, 1983

(k + 1)2sk+1 = (11k2 + 11k + 3)sk + k2sk−1, s0 = 1

z =∞�

k=0

skxk

x = q∞�

j=1

(1− q5j−4)5(1− q5j−1)5

(1− q5j−3)5(1− q5j−2)5= r5(q)

5

z =∞�

j=1

(1− qj)2

(1− q5j−4)5(1− q5j−1)5

1. Another proof that sk is an integer2. r5(q): the Rogers-Ramanujan continued fraction3. How to find other examples?

Sporadic sequences

Franel, 1894

(k + 1)2sk+1 = (7k2 + 7k + 2)sk + 8k2sk−1, s0 = 1

sk =k�

j=0

�k

j

�3

Apery, 1978

(k + 1)2sk+1 = (11k2 + 11k + 3)sk + k2sk−1, s0 = 1

sk =k�

j=0

�k

j

�2�k + j

j

�

Zagier, 1998, 2009

(k + 1)2sk+1 = (ak2 + ak + b)sk + ck2sk−1, s0 = 1


(a, b, c) s(k)

(11, 3, 1)�

j

�k

j

�2�k + j

j

�

(−17,−6,−72)�

j ,�

(−8)k−j

�k

j

��j

�

�3

(10, 3,−9)�

j

�k

j

�2�2jj

�

(7, 2, 8)�

j

�k

j

�3

(12, 4,−32)�

j

4k−2j

�k

2j

��2j

j

�2

(−9,−3,−27)�

j

(−3)k−3j

�k

j

��k − j

j

��k − 2j

j

�

Analogues of Beukers’ result

(k + 1)2sk+1 = (7k2 + 7k + 2)sk + 8k2sk−1, s0 = 1

z =∞�

k=0

skxk =

∞�

k=0

k�

j=0

�k

j

�3

xk

x = q∞�

j=1

(1− qj)3(1− q6j)9

(1− q2j)3(1− q3j)9= rc(q)

3

z =∞�

j=1

(1− q2j)(1− q3j)6

(1− qj)2(1− q6j)3

1. rc(q) is Ramanujan’s cubic continued fraction2. Similar results hold for Zagier’s other examples


(k + 1)2sk+1 = (11k2 + 11k + 3)sk + k2sk−1, s0 = 1

1. Three-term recurrence relation2. Coefficients are polynomials of degree 23. Zagier’s sporadic sequences



(k + 1)3tk+1 = (2k + 1)(17k2 + 17k + 5)tk − k3tk−1, t0 = 1

1. Three-term recurrence relation2. Coefficients are polynomials of degree 3

T. Sato, Abstract for Math. Soc. Japan, March 2002

(k + 1)3tk+1 = (2k + 1)(17k2 + 17k + 5)tk − k3tk−1, t0 = 1

x = q∞�

j=1

(1− qj)12(1− q6j)12

(1− q2j)12(1− q3j)12

(k+1)3tk+1 = −(2k+1)(11k2+11k+5)tk −125k3tk−1, t0 = 1

Analogues of

1

π=

2√2

9801

∞�

k=0

�2k

k

�2�4k2k

�(1103 + 26390k)

3964k.

H. H. Chan and coauthors, series of papers

Apery numbers and modular forms

f (5) :=5P(q5)− P(q)

4

=∞�

j=0

�2j

j

��j�

k=0

�j

k

�2�j + k

k

��η21η

25

f (5)

�2j

f (6) :=30P(q6)− 3P(q3) + 2P(q2)− 5P(q)

24

=∞�

j=0

�j�

k=0

�j

k

�2�j + k

k

�2��

η1η2η3η6f (6)

�2j

ηn = qn/24∞�

j=1

(1− qnj), P(q) = 1− 24∞�

j=1

jqj

1− qj

Outline





Jacobi, 1829

x =

∞�

n=−∞q(n+

12 )

2

∞�

n=−∞qn

2

4

, |q| < 1

Figure: Graph of x versus q

x(0) = 0

x(1) = 1dx

dq> 0

dx

dq

��q=0

= 16

Hypergeometric function

(a)n = a(a+ 1)(a+ 2) · · · (a+ n − 1), n ∈ Z+; (a)0 = 1

2F1(a, b; c ; x) =∞�

n=0

(a)n(b)n(c)nn!

xn

3F2(a, b, c ; d , e; x) =∞�

n=0

(a)n(b)n(c)n(d)n(e)nn!

xn

0F0(−;−; x) =∞�

n=0

1

n!xn

1F0(a;−; x) =∞�

n=0

(a)nn!

xn

Jacobi’s inversion formula, 1829

x =

∞�

n=−∞q(n+

12 )

2

∞�

n=−∞qn

2

4

, q = exp

�−π

2F1(12 ,

12 ; 1; 1− x)

2F1(12 ,

12 ; 1; x)

�


x(0) = 0

x(1) = 1

x(e−π) = 1/2

x(e−πt) + x(e−π/t) = 1, t > 0

Jacobi’s inversion formula, 1829

x =

∞�

n=−∞q(n+

12 )

2

∞�

n=−∞qn

2

4

, q = exp

�−π

2F1(12 ,

12 ; 1; 1− x)

2F1(12 ,

12 ; 1; x)

�


z = 2F1(1

2,1

2; 1; x) =

� ∞�

n=−∞qn

2

�2

qdx

dq= z2x(1− x)

Squares of hypergeometric functions

0F0 (−;−; x)2 = (ex)2 = 0F0 (−;−; 2x)

1F0 (a;−; x)2 =�(1− x)−a

�2= 1F0 (2a;−; x)

2F1 (a, b; c ; x)2 =

Clausen’s identity (1828)

Clausen’s identity

2F1

�1

2+ �,

1

2− �; 1; x

�2

= 3F2

�1

2+ �,

1

2− �,

1

2; 1, 1; 4x(1− x)

�

Clausen + Jacobi� ∞�

n=−∞qn

2

�4

= 3F2

�1

2,1

2,1

2; 1, 1; 4x(1− x)

�

After some manipulations (this expression generalizes)

f (4) :=4P(q4)− P(q)

3=

∞�

j=0

�2j

j

�3� η41η44

η42 f (4)

�2j

Aside: Jacobi’s sum of four squares theorem

� ∞�

n=−∞qn

2

�4

= 1 + 8∞�

j=14�j

jqj

1− qj

#�x21 + x22 + x23 + x24 = n, x1, x2, x3, x4 ∈ Z

�= 8

�

d|n4�d

d

Corollary (Lagrange) Every positive integer is a sum of foursquares.

#�x21 + x22 + x23 + x24 = n

�> 0

Another aside: Ramanujan (1914)

1

π=

1

16

∞�

j=0

�2j

j

�3 (42j + 5)

212j

Outline





Ramanujan (1914)There are similar theories when

q = exp

�−π

z(1− x)

z(x)

�, z(x) = 2F1

�1

2,1

2; 1; x

�

is replaced by any of

q1 = exp

�−π

√2z1(1− x)

z1(x)

�, z1(x) = 2F1

�1

4,3

4; 1; x

�

q2 = exp

�− 2π√

3

z2(1− x)

z2(x)

�, z2(x) = 2F1

�1

3,2

3; 1; x

�

q3 = exp

�−2π

z3(1− x)

z3(x)

�, z3(x) = 2F1

�1

6,5

6; 1; x

�

Ramanujan’s “alternative theories” of elliptic functions

Ramanujan (1914)

q = exp

�− 2π√

3

2F1�13 ,

23 ; 1; 1− x

�

2F1�13 ,

23 ; 1; x

��

J. M. Borwein and P. B. Borwein (1991)

x =

∞�

m=−∞

∞�

n=−∞q(m+1

3 )2+(m+1

3 )(n+13 )+(n+1

3 )2

∞�

m=−∞

∞�

n=−∞qm

2+mn+n2

3


q = exp

�− 2π√

3

2F1�13 ,

23 ; 1; 1− x

�

2F1�13 ,

23 ; 1; x

��


x(0) = 0

x(1) = 1

x(e−2π/√3) = 1/2


q = exp

�− 2π√

3

2F1�13 ,

23 ; 1; 1− x

�

2F1�13 ,

23 ; 1; x

��


z = 2F1

�1

3,2

3; 1; x

�

=∞�

m=−∞

∞�

n=−∞qm

2+mn+n2

qdx

dq= z2x(1− x)


Ramanujan

pp. 257–262, second notebook

27 Feb 1913, second letter to G. H. Hardy

1914 paper “Modular equations and approximations to 1/π”17 series for 1/π

Mordell (1927), Watson (1931)

“It is unfortunate that Ramanujan has not developed in detailthe corresponding theories...”

“There are developments of functions analogous to ellipticfunctions which I have not seen elsewhere...”

Fricke (1916)

Inversion formula for 2F1(16 ,

56 ; 1; x)


K. Venkatachaliengar (1988, republished 2012)

Initial investigations into the “alternative theories”

J. M. Borwein and P. B. Borwein (1987–1994)

A book and a series of papers

Proved all 17 of Ramanujan’s series for 1/π

Discovered the cubic theta function��

qm2+mn+n2

Berndt, Bhargava and Garvan (1995)

Proved all of the results on pp. 257–262 of Ramanujan’s secondnotebook. (Trans. Amer. Math. Soc., 82 pages)


H. H. Chan (1998)

The 2F1(13 ,

23 ; 1; x) theory

Berndt, Chan and Liaw (2001)

The 2F1(14 ,

34 ; 1; x) theory

K. S. Williams (2004)

The 2F1(13 ,

23 ; 1; x) theory

C., (2009)

A unified treatment for all four theories

D. Schultz (2013)

Cubic theory

Example of a modular form

Eisenstein series

E2n(τ) =�

(j ,k) �=(0,0)

1

(j + kτ)2n, n = 2, 3, . . . , Im(τ) > 0

Transformations

E2n(τ + 1) = E2n(τ)

E2n

�−1

τ

�= τ2nE2n(τ)

E2n is a modular form of weight 2n

E2n

�aτ + b

cτ + d

�= (cτ + d)2nE2n(τ)

for all integers a, b, c and d with ad − bc = 1.

The effect of scaling

Suppose

f

�aτ + b

cτ + d

�= (cτ + d)nf (τ)

for all integers a, b, c and d with ad − bc = 1.

Let m be a positive integer and let g(τ) = f (mτ). Then

g

�aτ + b

cτ + d

�= (cτ + d)ng(τ)

for all integers a, b, c and d with ad − bc = 1, provided inaddition c ≡ 0 (mod m).

Congruence subgroups

The modular group

Γ =

��a bc d

�� a, b, c , d ∈ Z, ad − bc = 1

�

Congruence subgroup

Γ0(m) =

��a bc d

�∈ Γ

�� c ≡ 0 (mod m)

�

Modular form

A function f is a modular form of weight n and level m if

f

�aτ + b

cτ + d

�= (cτ + d)nf (τ) for all

�a bc d

�∈ Γ0(m).

Examples of modular forms

Suppose q = exp(2πiτ) so Im(τ) > 0 ⇐⇒ |q| < 1

Let P(τ) = 1− 24∞�

n=1

nqn

1− qnQ(τ) = 1 + 240

∞�

n=1

n3qn

1− qn

P�

aτ+bcτ+d

�= (cτ + d)2P(τ)− πic(cτ + d).

P(τ) is not a modular form

mP(mτ)− P(τ) is a modular form of weight 2 and level m

Q(τ) is a modular form of weight 4 (and level 1)

Ramanujan’s alternative theories of elliptic functions

z41 = Q(τ)

z22 = 2P(2τ)− P(τ)

z23 =3P(3τ)− P(τ)

2

z24 =4P(4τ)− P(τ)

3

z1 = 2F1 ( 16 ,

56 ;1;x1)

z2 = 2F1 ( 14 ,

34 ;1;x2)

z3 = 2F1 ( 13 ,

23 ;1;x3)

z4 = 2F1 ( 12 ,

12 ;1;x4)

qdxmdq

= z2mxm(1− xm), xm(e−2π/

√m) =

1

2

A common way of viewing all 4 theories

f (4) :=4P(q4)− P(q)

3=

∞�

j=0

�2j

j

�3� η41η44

η42 f (4)

�2j

f (3) :=3P(q3)− P(q)

2=

∞�

j=0

�2j

j

�2�3jj

��η21η

23

f (3)

�3j

f (2) := 2P(q2)− P(q) =∞�

j=0

�2j

j

�2�4j2j

��η21η

22

f (2)

�4j

f (1) := Q(q)1/2 =∞�

j=0

�2j

j

��3j

j

��6j

3j

��η41f (1)

�6j

P(q) = 1− 24∞�

j=1

jqj

1− qj, Q(q) = 1 + 240

∞�

j=1

j3qj

1− qj

ηm = qm/24∞�

j=1

(1− qmj)

Outline





F. Beukers, 1983

(k + 1)2sk+1 = (11k2 + 11k + 3)sk + k2sk−1, s0 = 1

z =∞�

k=0

skxk

x = q∞�

j=1

(1− q5j−4)5(1− q5j−1)5

(1− q5j−3)5(1− q5j−2)5= r5(q)

5

z =∞�

j=1

(1− qj)2

(1− q5j−4)5(1− q5j−1)5

How does it fit in with Jacobi and Ramanujan’s theories?

Analogues of Clausen’s formula

Chan, Tanigawa, Yang and Zudilin (2011): Clausen 1

(1 + cw2)

∞�

j=0

ujwj

2

=∞�

j=0

�2j

j

�uj

�w(1− aw − cw2)

(1 + cw2)2

�j

Almkvist, van Straten and Zudilin (2011): Clausen 2

(1− aw − cw2)

∞�

j=0

ujwj

2

=∞�

j=0

tj

�w

1− aw − cw2

�j

(j + 1)2uj+1 = (aj2 + aj + b)uj + cj2uj−1

(j + 1)3tj+1 = −(2j + 1)(aj2 + aj + a− 2b)tj − (4c + a2)j3tj−1

u0 = t0 = 1

Higher levels: Clausen 1 example

f (4) :=4P(q4)− P(q)

3=

∞�

j=0

�2j

j

�3� η41η44

η42 f (4)

�2j

f (5) :=5P(q5)− P(q)

4

=∞�

j=0

�2j

j

��j�

k=0

�j

k

�2�j + k

k

��η21η

25

f (5)

�2j

sj =j�

k=0

�j

k

�2�j + k

k

�

(j + 1)2sj+1 = (11j2 + 11j + 3)sj + j2sj−1

R. Apery: ζ(2) �∈ Q

Rogers-Ramanujan continued fraction

5P(q5)− P(q)

4=

∞�

j=0

�2j

j

��j�

k=0

�j

k

�2�j + k

k

��η21η

25

f (5)

�2j

r = r(q) =q1/5

1 +q

1 +q2

1 +q3

1 + · · ·

.

�η21η

25

f (5)

�2

=r5(1− 11r5 − r10)

(1 + r10)2.

Higher levels: Clausen 2 example

f (4) :=4P(q4)− P(q)

3=

∞�

j=0

�2j

j

�3� η41η44

η42 f (4)

�2j

f (6) :=30P(q6)− 3P(q3) + 2P(q2)− 5P(q)

24

=∞�

j=0

�j�

k=0

�j

k

�2�j + k

k

�2��

η1η2η3η6f (6)

�2j

(j+1)3tj+1 = (2j+1)(17j2+17j+5)tj− j3tj−1 : Apery, ζ(3) �∈ Q

P(q) = 1− 24∞�

j=1

jqj

1− qj, ηm = qm/24

∞�

j=1

(1− qmj)

Summary, so far

Levels 1, 2, 3: Ramanujan’s theories to alternative bases

Level 4: Jacobi

Levels 5, 6, 6, 6, 8, 9: Zagier’s sporadic sequences

2F1 functions correspond to weight one modular forms.(e.g., elliptic integral ↔ sum of two squares)

(j + 1)2uj+1 = (aj2 + aj + b)uj + cj2uj−1

3F2 functions correspond to weight two modular forms.

(j + 1)3tj+1 = −(2j + 1)(aj2 + aj + a− 2b)tj − (4c + a2)j3tj−1

(j + 1)3sj+1 = 2(2j + 1)(aj2 + aj + b)sj + 4cj(4j2 − 1)sj−1

Other three-term recurrence relations

sj =j�

k=0

�j

k

�4

, level 10

(j + 1)3sj+1 = (2j + 1)(6j2 + 6j + 2)sj + j(64j2 − 4)sj−1, Franel

Experimental search:

(j + 1)3sj+1 = (2j + 1)(aj2 + aj + b)sj + j(cj2 + d)sj−1

(a, b, c , d) (13, 4, 27,−3) (6, 2, 64,−4) (14, 6,−192, 12)level 7 10 18

Level 7

f (7) =7P(q7)− P(q)

6=

∞�

j=−∞

∞�

k=−∞qj

2+jk+2k2

2

=∞�

j=0

�j/2��

k=0

�j

k

�2�2j − k

j

��2j − 2k

j

�

�η21η

27

f (7)

�3j/2

.

Level 10

f (10) :=10P(q10) + 5P(q5)− 2P(q2)− P(q)

12

=∞�

j=0

�j�

k=0

�j

k

�4��

η1η2η5η10f (10)

�4j/3

Level 13. Joint work with Dongxi Ye

(x1, x2, . . . , xm; q)∞ =∞�

j=0

(1− x1qj)(1− x2q

j) · · · (1− xmqj)

R = R(q) = q∞�

j=1

(1− qj)(j13) = q

(q, q3, q4, q9, q10, q12; q13)∞(q2, q5, q6, q7, q8, q11; q13)∞

= r5(q) = q∞�

j=1

(1− qj)5(j5) = q

(q, q4; q5)5∞(q2, q3; q5)5∞

1

R− 3− R =

1

q

∞�

j=1

(1− qj)2

(1− q13j)2

1 − 11− =1

q

∞�

j=1

(1− qj)6

(1− q5j)6

Factorizations: Rogers-Ramanujan continued fraction

1 − 11− =1

q

∞�

j=1

(1− qj)6

(1− q5j)6

1 − 11− =

�1√ − α5√

��1√ − β5√

�

1√ − α5√ =1

(q5s)1/12

∞�

j=1

1

(1− ζqj)5(1− ζ4qj)5

1√ − β5√ =1

(q5s)1/12

∞�

j=1

1

(1− ζ2qj)5(1− ζ3qj)5.

α =1−

√5

2, β =

1 +√5

2, ζ = exp(2πi/5), s =

η65η61

Factorizations: level 13

1

R− 3− R =

1

q

∞�

j=1

(1− qj)2

(1− q13j)2

1

R− 3− R =

�1√R

− γ√R

��1√R

− δ√R

�

1√R

− γ√R =

1

(qS)1/4× 1

(ξq, ξ3q, ξ4q, ξ9q, ξ10q, ξ12q; q)∞

1√R

− δ√R =

1

(qS)1/4× 1

(ξ2q, ξ5q, ξ6q, ξ7q, ξ8q, ξ11q; q)∞.

γ =3−

√13

2, δ =

3 +√13

2, ξ = exp(2πi/13), S =

η213η21

(q2, q5, q6, q7, q8, q11, q13, q13; q13)∞

−γ q (q, q3, q4, q9, q10, q12, q13, q13; q13)∞

= (ξ2q, ξ5q, ξ6q, ξ7q, ξ8q, ξ11q, q, q; q)∞

(q2, q5, q6, q7, q8, q11, q13, q13; q13)∞

−δ q (q, q3, q4, q9, q10, q12, q13, q13; q13)∞

= (ξq, ξ3q, ξ4q, ξ9q, ξ10q, ξ12q, q, q; q)∞.

γ =3−

√13

2, δ =

3 +√13

2, ξ = exp(2πi/13)

Weight 2 modular functions

qd

dqlog

�

1− 11 − 2

�=

∞�

n=0

a(n)

�(1− 11 − 2)

(1 + 2)2

�n

a(n) =

�2n

n

� n�

j=0

�n

j

�2�n + j

j

�.

The coefficients a(n) satisfy a 3-term recurrence relation.

qd

dqlog

�R

1− 3R − R2

�=

∞�

n=0

A(n)

�R(1− 3R − R2)

(1 + R2)2

�n

The coefficients A(n) satisfy a 6-term recurrence relation.

Degree 13 hypergeometric transformation formulas

2F1�

112 ,

512 ; 1;

1728x(1+5x+13x2)(1+247x+3380x2+15379x3+28561x4)3

�

4√1 + 247x + 3380x2 + 15379x3 + 28561x4

=2F1

�112 ,

512 ; 1;

1728x13

(1+5x+13x2)(1+7x+20x2+19x3+x4)3

�

4√1 + 7x + 20x2 + 19x3 + x4

3F2�16 ,

56 ,

12 ; 1, 1;

1728x(1+5x+13x2)(1+247x+3380x2+15379x3+28561x4)3

�

√1 + 247x + 3380x2 + 15379x3 + 28561x4

=3F2

�16 ,

56 ,

12 ; 1, 1;

1728x13

(1+5x+13x2)(1+7x+20x2+19x3+x4)3

�

√1 + 7x + 20x2 + 19x3 + x4

.

Other levels similar to 5 and 13: (�− 1) | 24

If (�− 1) | 24, let f (�) = �P(q�)− P(q)

�− 1.

f (�) =∞�

j=0

A�(j)xj�

where A�(j) satisfies a recurrence relation, of order given by:

� 2 3 4 5 7 9 13 25order 2 2 2 3 3 3 6 9

Level 11. Joint work with J. Ge and D. Ye

f (3) =3P(q3)− P(q)

2=

∞�

j=−∞

∞�

k=−∞qj

2+jk+k2

2

f (7) =7P(q7)− P(q)

6=

∞�

j=−∞

∞�

k=−∞qj

2+jk+2k2

2

f (11) =

∞�

j=−∞

∞�

k=−∞qj

2+jk+3k2

2

f (�) =∞�

j=0

A�(j)

�η21η

2�

f (�)

�12j/(�+1)

, � = 3, 7, 11

(j + 1)3A11(j + 1) = 2(2j + 1)(5j2 + 5j + 2)A11(j)

−8j(7j2 + 1)A11(j − 1) + 22j(j − 1)(2j − 1)A11(j − 2).

Levels 14 and 15. Joint work with Dongxi Ye

Similar results, because:

�

d |14

d = 1 + 2 + 7 + 14 = 24�

d |15

d = 1 + 3 + 5 + 15 = 24

Cubic transformation of a level 5 function

f (x) =∞�

n=0

�2n

n

�

n�

j=0

�n

j

�2�n + j

j

�

xn =∞�

n=0

cnxn

1

1 + 9x + 27x2f

�x

(1 + 9x + 27x2)2

�

=1

1 + 3x + 3x2f

�x3

(1 + 3x + 3x2)2

�

Quintic transformation of a level 3 function

g(x) =∞�

n=0

anxn, a0 = 1

(n + 1)3an+1 = (2n + 1)(7n2 + 7n + 3)an

− n(29n2 + 4)an−1 + 30n(n − 1)(2n − 1)an−2

x =v

(1 + 3v)2=

w

1 + w − w2near x = 0

xg(x) =vw

3v(1 + w2)− 2w(1− 9v2) 3F2

� 13 ,

12 ,

23

1, 1;

108v3w2

(w2 + 27v3)2

�

=5vw

3v(1 + w2) + 2w(1− 9v2) 3F2

� 13 ,

12 ,

23

1, 1;

108v3w2

(1 + 27v3w2)2

�.

Jacobi, level 4

∞�

j=−∞qj

2

4

=

∞�

j=−∞(−1)jqj

2

4

+

∞�

j=−∞q(j+

12 )

2

4

Level 15 analogue

∞�

j=−∞

∞�

k=−∞qj

2+jk+k2

2

+ 5

∞�

j=−∞

∞�

k=−∞q5j

2+5jk+5k2

2

= 3

∞�

j=−∞

∞�

k=−∞qj

2+jk+4k2

2

+ 3

∞�

j=−∞

∞�

k=−∞q2j

2+jk+2k2

2

Other levels

Level 10

t(k) =k�

j=0

�k

j

�4

, Franel, 1895, four-term recurrence

κ = r(q)r2(q2) = q∞�

j=1

(1− q10j−9)(1− q10j−8)(1− q10j−2)(1− q10j−1)

(1− q10j−7)(1− q10j−6)(1− q10j−4)(1− q10j−3)

Level 12

κ12 = q∞�

j=1

(1− q12j−11)(1− q12j−1)

(1− q12j−5)(1− q12j−7)

The coefficients satisfy a six-term recurrence relation.

There are also results for levels 18, 20, 24, 32

Summary

Table: Order of recurrence relations for level �

� 1..4 5..10 11 12, 13 14..16 18 20, 24, 32 25order 2 3 4 6 4 3 4 9

Levels 1, 2, 3: Ramanujan’s “alternative theories”

Level 4: Jacobi

Levels 5, 6, 8, 9: Zagier’s sporadic sequences

Level 5: Rogers-Ramanujan cont. frac. Weight 1: Apery ζ(2)

Level 6: 3 theories, one involves the cubic continued fraction

Weight 2: Apery ζ(3), Domb, Almkvist-Zudilin numbers

Levels 2, 3, 4, 5, 7, 9, 13, 25: (�− 1)|24 share a common theory

Levels 3, 7, 11: another common theory

Levels 14, 15: very similar theories

Ramanujan’s series for 1/π

Ramanujan-Gosper, level 2, degree 29

1

π=

2√2

9801

∞�

k=0

�2k

k

�2�4k2k

�(1103 + 26390k)

3964k.

�η21η

22

2P(q2)− P(q)

�4��q=exp(−2π

√29/2)

=1

3964

Another example, level 11, degree 17

1

π=

√11

484

∞�

k=0

(−1)k c(k)(67 + 221k)

44k

where c(k) satisfies a 4-term recurrence relation.

�η1η11�

j

�k q

j2+jk+3k2

�2��q=− exp(−π

√17/11)

=−1

44

Another example, with an “80” in it

Zagier’s sporadic sequence: (a, b, c) = (10, 3,−9)

Level: � = 6

Degree: N = 5

1

π=

1

37/2

∞�

k=0

�2k

k

�

k�

j=0

�k

j

�2�2jj

�

(13 + 80k)

182k

x :=

�4η1η2η3η6

6P6 − 3P3 + 2P2 − P1

�2��q=exp(−2π

√N/�)

=1

182

�N

�× 2

�1− 4ax − 16cx2 =

80

37/2.

The end!

Ramanujan’s theories of elliptic functions to alternative...

Documents

Transcript of Ramanujan’s theories of elliptic functions to alternative...