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.

Apery’s numbers used to prove ζ(3) �∈ Q

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

The numbers tk are integers—not obvious.

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

(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

(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

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

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

Apery’s numbers used to prove ζ(3) �∈ Q

(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

1 Sporadic sequences

2 Background: classical theory

3 Ramanujan’s alternative theories

4 Beyond Ramanujan’s alternative theories

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)

�

Figure: Graph of x versus q

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)

�

Figure: Graph of x versus q

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

1 Sporadic sequences

2 Background: classical theory

3 Ramanujan’s alternative theories

4 Beyond Ramanujan’s alternative theories

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

Ramanujan’s “alternative theories” of elliptic functions

q = exp

�− 2π√

3

2F1�13 ,

23 ; 1; 1− x

�

2F1�13 ,

23 ; 1; x

��

Figure: Graph of x versus q

x(0) = 0

x(1) = 1

x(e−2π/√3) = 1/2

Ramanujan’s “alternative theories” of elliptic functions

q = exp

�− 2π√

3

2F1�13 ,

23 ; 1; 1− x

�

2F1�13 ,

23 ; 1; x

��

Figure: Graph of x versus q

z = 2F1

�1

3,2

3; 1; x

�

=∞�

m=−∞

∞�

n=−∞qm

2+mn+n2

qdx

dq= z2x(1− x)

Ramanujan’s “alternative theories” of elliptic functions

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)

Ramanujan’s “alternative theories” of elliptic functions

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)

Ramanujan’s “alternative theories” of elliptic functions

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

1 Sporadic sequences

2 Background: classical theory

3 Ramanujan’s alternative theories

4 Beyond Ramanujan’s alternative theories

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!