Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev...

126
Ramanujan’s series for 1Shaun Cooper Massey University, Auckland, New Zealand

Transcript of Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev...

Page 1: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Ramanujan’s series for 1/π

Shaun Cooper

Massey University, Auckland, New Zealand

Page 2: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Srinivasa Ramanujan, 1887 – 1920

Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in“The Man who knew Infinity”, based on a book by Robert Kanigel.

Page 3: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a
Page 4: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a
Page 5: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a
Page 6: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Goals of today’s lecture:

1. To outline a general theory that accounts for all of these series.

2. To sketch the ideas of a proof, with specific reference to theseries

∞∑j=0

(2j

j

)3(j +

1

6

)(1

256

)j

=2

3× 1

π.

3. To indicate some directions for further research.

Page 7: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Goals of today’s lecture:

1. To outline a general theory that accounts for all of these series.

2. To sketch the ideas of a proof, with specific reference to theseries

∞∑j=0

(2j

j

)3(j +

1

6

)(1

256

)j

=2

3× 1

π.

3. To indicate some directions for further research.

Page 8: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Goals of today’s lecture:

1. To outline a general theory that accounts for all of these series.

2. To sketch the ideas of a proof, with specific reference to theseries

∞∑j=0

(2j

j

)3(j +

1

6

)(1

256

)j

=2

3× 1

π.

3. To indicate some directions for further research.

Page 9: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Goals of today’s lecture:

1. To outline a general theory that accounts for all of these series.

2. To sketch the ideas of a proof, with specific reference to theseries

∞∑j=0

(2j

j

)3(j +

1

6

)(1

256

)j

=2

3× 1

π.

3. To indicate some directions for further research.

Page 10: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Ramanujan’s series rely on four hypotheses.

Hypothesis 1. There is a positive integer `, called the level, suchthat the transformation formulas

t Z(e−2π√

t/`)

= Z(e−2π/

√t`)

andX(e−2π√

t/`)

= X(e−2π/

√t`)

hold for all values of a positive real variable t.

For the next slide, recall that

P(q) = 1− 24∞∑j=1

jqj

1− qjand ηm = qm/24

∞∏j=1

(1− qmj).

Page 11: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Ramanujan’s series rely on four hypotheses.

Hypothesis 1. There is a positive integer `, called the level, suchthat the transformation formulas

t Z(e−2π√

t/`)

= Z(e−2π/

√t`)

andX(e−2π√

t/`)

= X(e−2π/

√t`)

hold for all values of a positive real variable t.

For the next slide, recall that

P(q) = 1− 24∞∑j=1

jqj

1− qjand ηm = qm/24

∞∏j=1

(1− qmj).

Page 12: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Ramanujan’s series rely on four hypotheses.

Hypothesis 1. There is a positive integer `, called the level, suchthat the transformation formulas

t Z(e−2π√

t/`)

= Z(e−2π/

√t`)

andX(e−2π√

t/`)

= X(e−2π/

√t`)

hold for all values of a positive real variable t.

For the next slide, recall that

P(q) = 1− 24∞∑j=1

jqj

1− qjand ηm = qm/24

∞∏j=1

(1− qmj).

Page 13: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Z4 = 13

(4P(q4)− P(q)

), X4 =

(η41η

44

η42 Z4

)2

, ` = 4

Z3 = 12

(3P(q3)− P(q)

), X3

(η21η

23

Z3

)3

, ` = 3

Z2 = 2P(q2)− P(q), X2 =

(η21η

22

Z2

)4

, ` = 2

Z1 = Q(q)1/2, X1 =

(η41Z1

)6

, ` = 1

Z10 =1

12

(10P(q10 + 5P(q5)− 2P(q2)− P(q)

)X10 =

(η1η2η5η10

Z10

)4/3

, ` = 10

Page 14: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Z4 = 13

(4P(q4)− P(q)

), X4 =

(η41η

44

η42 Z4

)2

, ` = 4

Z3 = 12

(3P(q3)− P(q)

), X3

(η21η

23

Z3

)3

, ` = 3

Z2 = 2P(q2)− P(q), X2 =

(η21η

22

Z2

)4

, ` = 2

Z1 = Q(q)1/2, X1 =

(η41Z1

)6

, ` = 1

Z10 =1

12

(10P(q10 + 5P(q5)− 2P(q2)− P(q)

)X10 =

(η1η2η5η10

Z10

)4/3

, ` = 10

Page 15: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Z4 = 13

(4P(q4)− P(q)

), X4 =

(η41η

44

η42 Z4

)2

, ` = 4

Z3 = 12

(3P(q3)− P(q)

), X3

(η21η

23

Z3

)3

, ` = 3

Z2 = 2P(q2)− P(q), X2 =

(η21η

22

Z2

)4

, ` = 2

Z1 = Q(q)1/2, X1 =

(η41Z1

)6

, ` = 1

Z10 =1

12

(10P(q10 + 5P(q5)− 2P(q2)− P(q)

)X10 =

(η1η2η5η10

Z10

)4/3

, ` = 10

Page 16: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Z4 = 13

(4P(q4)− P(q)

), X4 =

(η41η

44

η42 Z4

)2

, ` = 4

Z3 = 12

(3P(q3)− P(q)

), X3

(η21η

23

Z3

)3

, ` = 3

Z2 = 2P(q2)− P(q), X2 =

(η21η

22

Z2

)4

, ` = 2

Z1 = Q(q)1/2, X1 =

(η41Z1

)6

, ` = 1

Z10 =1

12

(10P(q10 + 5P(q5)− 2P(q2)− P(q)

)X10 =

(η1η2η5η10

Z10

)4/3

, ` = 10

Page 17: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Z4 = 13

(4P(q4)− P(q)

), X4 =

(η41η

44

η42 Z4

)2

, ` = 4

Z3 = 12

(3P(q3)− P(q)

), X3

(η21η

23

Z3

)3

, ` = 3

Z2 = 2P(q2)− P(q), X2 =

(η21η

22

Z2

)4

, ` = 2

Z1 = Q(q)1/2, X1 =

(η41Z1

)6

, ` = 1

Z10 =1

12

(10P(q10 + 5P(q5)− 2P(q2)− P(q)

)X10 =

(η1η2η5η10

Z10

)4/3

, ` = 10

Page 18: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Hypothesis 2. There is a power series expansion

Z =∞∑j=0

h(j)X j

that converges in a neighborhood of X = 0.

Z4 =1

3

(4P(q4)− P(q)

)= 1+8q+24q2+32q3+24q4+48q5+· · ·

X4 =

(η41η

44

η42 Z4

)2

= q − 24q2 + 296q3 − 2528q4 + 16928q5 + · · ·

Z4 = 1 + 8X4 + 216X 24 + 8032X 3

4 + · · ·

In fact,

Z4 =∞∑j=0

(2j

j

)3

X j4, so h(j) =

(2j

j

)3

.

Page 19: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Hypothesis 2. There is a power series expansion

Z =∞∑j=0

h(j)X j

that converges in a neighborhood of X = 0.

Z4 =1

3

(4P(q4)− P(q)

)= 1+8q+24q2+32q3+24q4+48q5+· · ·

X4 =

(η41η

44

η42 Z4

)2

= q − 24q2 + 296q3 − 2528q4 + 16928q5 + · · ·

Z4 = 1 + 8X4 + 216X 24 + 8032X 3

4 + · · ·

In fact,

Z4 =∞∑j=0

(2j

j

)3

X j4, so h(j) =

(2j

j

)3

.

Page 20: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Hypothesis 2. There is a power series expansion

Z =∞∑j=0

h(j)X j

that converges in a neighborhood of X = 0.

Z4 =1

3

(4P(q4)− P(q)

)= 1+8q+24q2+32q3+24q4+48q5+· · ·

X4 =

(η41η

44

η42 Z4

)2

= q − 24q2 + 296q3 − 2528q4 + 16928q5 + · · ·

Z4 = 1 + 8X4 + 216X 24 + 8032X 3

4 + · · ·

In fact,

Z4 =∞∑j=0

(2j

j

)3

X j4, so h(j) =

(2j

j

)3

.

Page 21: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Hypothesis 2. There is a power series expansion

Z =∞∑j=0

h(j)X j

that converges in a neighborhood of X = 0.

Z4 =1

3

(4P(q4)− P(q)

)= 1+8q+24q2+32q3+24q4+48q5+· · ·

X4 =

(η41η

44

η42 Z4

)2

= q − 24q2 + 296q3 − 2528q4 + 16928q5 + · · ·

Z4 = 1 + 8X4 + 216X 24 + 8032X 3

4 + · · ·

In fact,

Z4 =∞∑j=0

(2j

j

)3

X j4, so h(j) =

(2j

j

)3

.

Page 22: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Hypothesis 2. Further examples:

Z4 =∞∑j=0

(2j

j

)3

X j4,

Z3 =∞∑j=0

(4j

2j

)(2j

j

)2

X j3,

Z2 =∞∑j=0

(3j

j

)(2j

j

)2

X j2,

Z1 =∞∑j=0

(6j

3j

)(3j

j

)(2j

j

)X j1,

Z10 =∞∑j=0

h(j)X j10, h(j) =

j∑i=0

(j

i

)4

Page 23: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Hypothesis 2. Further examples:

Z4 =∞∑j=0

(2j

j

)3

X j4,

Z3 =∞∑j=0

(4j

2j

)(2j

j

)2

X j3,

Z2 =∞∑j=0

(3j

j

)(2j

j

)2

X j2,

Z1 =∞∑j=0

(6j

3j

)(3j

j

)(2j

j

)X j1,

Z10 =∞∑j=0

h(j)X j10, h(j) =

j∑i=0

(j

i

)4

Page 24: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Hypothesis 2. Further examples:

Z4 =∞∑j=0

(2j

j

)3

X j4,

Z3 =∞∑j=0

(4j

2j

)(2j

j

)2

X j3,

Z2 =∞∑j=0

(3j

j

)(2j

j

)2

X j2,

Z1 =∞∑j=0

(6j

3j

)(3j

j

)(2j

j

)X j1,

Z10 =∞∑j=0

h(j)X j10, h(j) =

j∑i=0

(j

i

)4

Page 25: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Hypothesis 2. Further examples:

Z4 =∞∑j=0

(2j

j

)3

X j4,

Z3 =∞∑j=0

(4j

2j

)(2j

j

)2

X j3,

Z2 =∞∑j=0

(3j

j

)(2j

j

)2

X j2,

Z1 =∞∑j=0

(6j

3j

)(3j

j

)(2j

j

)X j1,

Z10 =∞∑j=0

h(j)X j10, h(j) =

j∑i=0

(j

i

)4

Page 26: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Hypothesis 2. Further examples:

Z4 =∞∑j=0

(2j

j

)3

X j4,

Z3 =∞∑j=0

(4j

2j

)(2j

j

)2

X j3,

Z2 =∞∑j=0

(3j

j

)(2j

j

)2

X j2,

Z1 =∞∑j=0

(6j

3j

)(3j

j

)(2j

j

)X j1,

Z10 =∞∑j=0

h(j)X j10, h(j) =

j∑i=0

(j

i

)4

Page 27: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Hypothesis 3. There is an algebraic function B = B(X ) such thatthe differentiation formula

qd

dqlogX = Z B(X )

holds.

Example: In Lecture 3 we proved (for the level 4 theory)

qdx

dq= z2x(1− x).

Under the change of variables Z4 = z2 and X4 = x(1− x)/16 thisbecomes

qdX

dq= Z√

1− 64X , so B(X ) =√

1− 64X .

Page 28: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Hypothesis 3. There is an algebraic function B = B(X ) such thatthe differentiation formula

qd

dqlogX = Z B(X )

holds.

Example: In Lecture 3 we proved (for the level 4 theory)

qdx

dq= z2x(1− x).

Under the change of variables Z4 = z2 and X4 = x(1− x)/16 thisbecomes

qdX

dq= Z√

1− 64X , so B(X ) =√

1− 64X .

Page 29: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Hypothesis 3. There is an algebraic function B = B(X ) such thatthe differentiation formula

qd

dqlogX = Z B(X )

holds.

Example: In Lecture 3 we proved (for the level 4 theory)

qdx

dq= z2x(1− x).

Under the change of variables Z4 = z2 and X4 = x(1− x)/16 thisbecomes

qdX

dq= Z√

1− 64X , so B(X ) =√

1− 64X .

Page 30: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Hypothesis 3.

qd

dqlogX = Z B(X )

` = 4 B =√

1− 64X

` = 3 B =√

1− 108X

` = 2 B =√

1− 256X

` = 1 B =√

1− 1728X

` = 10 B =√

(1− 16X )(1 + 4X )

Page 31: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Hypothesis 3.

qd

dqlogX = Z B(X )

` = 4 B =√

1− 64X

` = 3 B =√

1− 108X

` = 2 B =√

1− 256X

` = 1 B =√

1− 1728X

` = 10 B =√

(1− 16X )(1 + 4X )

Page 32: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Hypothesis 3.

qd

dqlogX = Z B(X )

` = 4 B =√

1− 64X

` = 3 B =√

1− 108X

` = 2 B =√

1− 256X

` = 1 B =√

1− 1728X

` = 10 B =√

(1− 16X )(1 + 4X )

Page 33: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Hypothesis 3.

qd

dqlogX = Z B(X )

` = 4 B =√

1− 64X

` = 3 B =√

1− 108X

` = 2 B =√

1− 256X

` = 1 B =√

1− 1728X

` = 10 B =√

(1− 16X )(1 + 4X )

Page 34: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Hypothesis 3.

qd

dqlogX = Z B(X )

` = 4 B =√

1− 64X

` = 3 B =√

1− 108X

` = 2 B =√

1− 256X

` = 1 B =√

1− 1728X

` = 10 B =√

(1− 16X )(1 + 4X )

Page 35: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Hypothesis 4.

Fix an integer N ≥ 2, called the degree. Let Y = Y (q) = X (qN).

The functions X and Y are algebraically dependent. That is, thereexists an irreducible polynomial g(X ,Y ), not identically zero, suchthat g(X ,Y ) = 0. The equation g(X ,Y ) = 0 is said to be amodular equation of degree N.

Example: level ` = 4, degree N = 3

X := X4 =

(η41η

44

η42 Z4

)2

= q−24q2+296q3−2528q4+16928q5+· · ·

Y := Y4 = X4(q3) = q3−24q6+296q9−2528q12+16928q15+ · · ·

X 4+Y 4−16777216X 3Y 3+294912 (X 3Y 2+X 2Y 3)−900(X 3Y+XY 3)

+28422X 2Y 2 + 72(X 2Y + XY 2)− XY = 0

Page 36: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Hypothesis 4.

Fix an integer N ≥ 2, called the degree. Let Y = Y (q) = X (qN).

The functions X and Y are algebraically dependent.

That is, thereexists an irreducible polynomial g(X ,Y ), not identically zero, suchthat g(X ,Y ) = 0. The equation g(X ,Y ) = 0 is said to be amodular equation of degree N.

Example: level ` = 4, degree N = 3

X := X4 =

(η41η

44

η42 Z4

)2

= q−24q2+296q3−2528q4+16928q5+· · ·

Y := Y4 = X4(q3) = q3−24q6+296q9−2528q12+16928q15+ · · ·

X 4+Y 4−16777216X 3Y 3+294912 (X 3Y 2+X 2Y 3)−900(X 3Y+XY 3)

+28422X 2Y 2 + 72(X 2Y + XY 2)− XY = 0

Page 37: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Hypothesis 4.

Fix an integer N ≥ 2, called the degree. Let Y = Y (q) = X (qN).

The functions X and Y are algebraically dependent. That is, thereexists an irreducible polynomial g(X ,Y ), not identically zero, suchthat g(X ,Y ) = 0.

The equation g(X ,Y ) = 0 is said to be amodular equation of degree N.

Example: level ` = 4, degree N = 3

X := X4 =

(η41η

44

η42 Z4

)2

= q−24q2+296q3−2528q4+16928q5+· · ·

Y := Y4 = X4(q3) = q3−24q6+296q9−2528q12+16928q15+ · · ·

X 4+Y 4−16777216X 3Y 3+294912 (X 3Y 2+X 2Y 3)−900(X 3Y+XY 3)

+28422X 2Y 2 + 72(X 2Y + XY 2)− XY = 0

Page 38: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Hypothesis 4.

Fix an integer N ≥ 2, called the degree. Let Y = Y (q) = X (qN).

The functions X and Y are algebraically dependent. That is, thereexists an irreducible polynomial g(X ,Y ), not identically zero, suchthat g(X ,Y ) = 0. The equation g(X ,Y ) = 0 is said to be amodular equation of degree N.

Example: level ` = 4, degree N = 3

X := X4 =

(η41η

44

η42 Z4

)2

= q−24q2+296q3−2528q4+16928q5+· · ·

Y := Y4 = X4(q3) = q3−24q6+296q9−2528q12+16928q15+ · · ·

X 4+Y 4−16777216X 3Y 3+294912 (X 3Y 2+X 2Y 3)−900(X 3Y+XY 3)

+28422X 2Y 2 + 72(X 2Y + XY 2)− XY = 0

Page 39: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Hypothesis 4.

Fix an integer N ≥ 2, called the degree. Let Y = Y (q) = X (qN).

The functions X and Y are algebraically dependent. That is, thereexists an irreducible polynomial g(X ,Y ), not identically zero, suchthat g(X ,Y ) = 0. The equation g(X ,Y ) = 0 is said to be amodular equation of degree N.

Example: level ` = 4, degree N = 3

X := X4 =

(η41η

44

η42 Z4

)2

= q−24q2+296q3−2528q4+16928q5+· · ·

Y := Y4 = X4(q3) = q3−24q6+296q9−2528q12+16928q15+ · · ·

X 4+Y 4−16777216X 3Y 3+294912 (X 3Y 2+X 2Y 3)−900(X 3Y+XY 3)

+28422X 2Y 2 + 72(X 2Y + XY 2)− XY = 0

Page 40: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Hypothesis 4.

Fix an integer N ≥ 2, called the degree. Let Y = Y (q) = X (qN).

The functions X and Y are algebraically dependent. That is, thereexists an irreducible polynomial g(X ,Y ), not identically zero, suchthat g(X ,Y ) = 0. The equation g(X ,Y ) = 0 is said to be amodular equation of degree N.

Example: level ` = 4, degree N = 3

X := X4 =

(η41η

44

η42 Z4

)2

= q−24q2+296q3−2528q4+16928q5+· · ·

Y := Y4 = X4(q3) = q3−24q6+296q9−2528q12+16928q15+ · · ·

X 4+Y 4−16777216X 3Y 3+294912 (X 3Y 2+X 2Y 3)−900(X 3Y+XY 3)

+28422X 2Y 2 + 72(X 2Y + XY 2)− XY = 0

Page 41: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Hypothesis 4.

Fix an integer N ≥ 2, called the degree. Let Y = Y (q) = X (qN).

The functions X and Y are algebraically dependent. That is, thereexists an irreducible polynomial g(X ,Y ), not identically zero, suchthat g(X ,Y ) = 0. The equation g(X ,Y ) = 0 is said to be amodular equation of degree N.

Example: level ` = 4, degree N = 3

X := X4 =

(η41η

44

η42 Z4

)2

= q−24q2+296q3−2528q4+16928q5+· · ·

Y := Y4 = X4(q3) = q3−24q6+296q9−2528q12+16928q15+ · · ·

X 4+Y 4−16777216X 3Y 3+294912 (X 3Y 2+X 2Y 3)−900(X 3Y+XY 3)

+28422X 2Y 2 + 72(X 2Y + XY 2)− XY = 0

Page 42: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Hypothesis 4.

Fix an integer N ≥ 2, called the degree. Let Y = Y (q) = X (qN).

The functions X and Y are algebraically dependent. That is, thereexists an irreducible polynomial g(X ,Y ), not identically zero, suchthat g(X ,Y ) = 0. The equation g(X ,Y ) = 0 is said to be amodular equation of degree N.

Example: level ` = 4, degree N = 3

X := X4 =

(η41η

44

η42 Z4

)2

= q−24q2+296q3−2528q4+16928q5+· · ·

Y := Y4 = X4(q3) = q3−24q6+296q9−2528q12+16928q15+ · · ·

X 4+Y 4−16777216X 3Y 3+294912 (X 3Y 2+X 2Y 3)−900(X 3Y+XY 3)

+28422X 2Y 2 + 72(X 2Y + XY 2)− XY = 0

Page 43: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Let X , Y , Z , B, h(j), ` and N be as for Hypotheses 1–4.

Let λ = λ(q) be defined by

λ =X

2

d

dX

(Y B(Y )

X B(X )÷ dY

dX

),

where the derivatives may be computed from the modular equationby implicit differentiation.Let X`,N , B`,N and λ`,N be defined by

X`,N = X(e−2π√

N/`), B`,N = B (X`,N)

andλ`,N = λ

(e−2π/

√N`).

Then

∞∑j=0

h(j) (j + λ`,N) (X`,N)j =1

2π× 1

B`,N×√`

N.

Page 44: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Let X , Y , Z , B, h(j), ` and N be as for Hypotheses 1–4.Let λ = λ(q) be defined by

λ =X

2

d

dX

(Y B(Y )

X B(X )÷ dY

dX

),

where the derivatives may be computed from the modular equationby implicit differentiation.

Let X`,N , B`,N and λ`,N be defined by

X`,N = X(e−2π√

N/`), B`,N = B (X`,N)

andλ`,N = λ

(e−2π/

√N`).

Then

∞∑j=0

h(j) (j + λ`,N) (X`,N)j =1

2π× 1

B`,N×√`

N.

Page 45: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Let X , Y , Z , B, h(j), ` and N be as for Hypotheses 1–4.Let λ = λ(q) be defined by

λ =X

2

d

dX

(Y B(Y )

X B(X )÷ dY

dX

),

where the derivatives may be computed from the modular equationby implicit differentiation.Let X`,N , B`,N and λ`,N be defined by

X`,N = X(e−2π√

N/`), B`,N = B (X`,N)

andλ`,N = λ

(e−2π/

√N`).

Then

∞∑j=0

h(j) (j + λ`,N) (X`,N)j =1

2π× 1

B`,N×√`

N.

Page 46: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Let X , Y , Z , B, h(j), ` and N be as for Hypotheses 1–4.Let λ = λ(q) be defined by

λ =X

2

d

dX

(Y B(Y )

X B(X )÷ dY

dX

),

where the derivatives may be computed from the modular equationby implicit differentiation.Let X`,N , B`,N and λ`,N be defined by

X`,N = X(e−2π√

N/`), B`,N = B (X`,N)

andλ`,N = λ

(e−2π/

√N`).

Then

∞∑j=0

h(j) (j + λ`,N) (X`,N)j =1

2π× 1

B`,N×√`

N.

Page 47: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Example:

∞∑j=0

(2j

j

)3(j +

1

6

)(1

256

)j

=2

3× 1

π.

Here, the level is 4 and degree is 3.

By Hypothesis 2 with ` = 4, we have

Z4 =1

3

(4P(q4)− P(q)

), X4 =

(η41η

44

η42 Z4

)2

Z4 =∞∑j=0

(2j

j

)3

X j4, so h(j) =

(2j

j

)3

.

Page 48: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Example:

∞∑j=0

(2j

j

)3(j +

1

6

)(1

256

)j

=2

3× 1

π.

Here, the level is 4 and degree is 3.

By Hypothesis 2 with ` = 4, we have

Z4 =1

3

(4P(q4)− P(q)

), X4 =

(η41η

44

η42 Z4

)2

Z4 =∞∑j=0

(2j

j

)3

X j4, so h(j) =

(2j

j

)3

.

Page 49: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Example:

∞∑j=0

(2j

j

)3(j +

1

6

)(1

256

)j

=2

3× 1

π.

Here, the level is 4 and degree is 3.

By Hypothesis 2 with ` = 4, we have

Z4 =1

3

(4P(q4)− P(q)

), X4 =

(η41η

44

η42 Z4

)2

Z4 =∞∑j=0

(2j

j

)3

X j4, so h(j) =

(2j

j

)3

.

Page 50: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

The degree is N = 3. With Y (q) = X (q3) we have (Hypothesis 4)

X 4+Y 4−16777216X 3Y 3+294912 (X 3Y 2+X 2Y 3)−900(X 3Y+XY 3)

+28422X 2Y 2 + 72(X 2Y + XY 2)− XY = 0

where

X = q∞∏j=1

(1− qj)24(1− q4j)24

(1− q2j)48.

Page 51: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

The degree is N = 3. With Y (q) = X (q3) we have (Hypothesis 4)

X 4+Y 4−16777216X 3Y 3+294912 (X 3Y 2+X 2Y 3)−900(X 3Y+XY 3)

+28422X 2Y 2 + 72(X 2Y + XY 2)− XY = 0

When q = e−2π√

N/` = e−π/√3, Hypothesis 1 implies X = Y .

The modular equation reduces to

X 2(1− 256X )(1− 16X )(1 + 64X 2) = 0.

A numerical approximation determines that X (e−π/√3) = 1

256 .

∞∑j=0

(2j

j

)3(j +

1

6

)(1

256

)j

=2

3× 1

π.

Page 52: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

The degree is N = 3. With Y (q) = X (q3) we have (Hypothesis 4)

X 4+Y 4−16777216X 3Y 3+294912 (X 3Y 2+X 2Y 3)−900(X 3Y+XY 3)

+28422X 2Y 2 + 72(X 2Y + XY 2)− XY = 0

When q = e−2π√

N/` = e−π/√3, Hypothesis 1 implies X = Y .

The modular equation reduces to

X 2(1− 256X )(1− 16X )(1 + 64X 2) = 0.

A numerical approximation determines that X (e−π/√3) = 1

256 .

∞∑j=0

(2j

j

)3(j +

1

6

)(1

256

)j

=2

3× 1

π.

Page 53: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

The degree is N = 3. With Y (q) = X (q3) we have (Hypothesis 4)

X 4+Y 4−16777216X 3Y 3+294912 (X 3Y 2+X 2Y 3)−900(X 3Y+XY 3)

+28422X 2Y 2 + 72(X 2Y + XY 2)− XY = 0

When q = e−2π√

N/` = e−π/√3, Hypothesis 1 implies X = Y .

The modular equation reduces to

X 2(1− 256X )(1− 16X )(1 + 64X 2) = 0.

A numerical approximation determines that X (e−π/√3) = 1

256 .

∞∑j=0

(2j

j

)3(j +

1

6

)(1

256

)j

=2

3× 1

π.

Page 54: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

The degree is N = 3. With Y (q) = X (q3) we have (Hypothesis 4)

X 4+Y 4−16777216X 3Y 3+294912 (X 3Y 2+X 2Y 3)−900(X 3Y+XY 3)

+28422X 2Y 2 + 72(X 2Y + XY 2)− XY = 0

When q = e−2π√

N/` = e−π/√3, Hypothesis 1 implies X = Y .

The modular equation reduces to

X 2(1− 256X )(1− 16X )(1 + 64X 2) = 0.

A numerical approximation determines that X (e−π/√3) = 1

256 .

∞∑j=0

(2j

j

)3(j +

1

6

)(1

256

)j

=2

3× 1

π.

Page 55: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

The degree is N = 3. With Y (q) = X (q3) we have (Hypothesis 4)

X 4+Y 4−16777216X 3Y 3+294912 (X 3Y 2+X 2Y 3)−900(X 3Y+XY 3)

+28422X 2Y 2 + 72(X 2Y + XY 2)− XY = 0

When q = e−2π√

N/` = e−π/√3, Hypothesis 1 implies X = Y .

The modular equation reduces to

X 2(1− 256X )(1− 16X )(1 + 64X 2) = 0.

A numerical approximation determines that X (e−π/√3) = 1

256 .

∞∑j=0

(2j

j

)3(j +

1

6

)(1

256

)j

=2

3× 1

π.

Page 56: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

∞∑j=0

h(j) (j + λ`,N) (X`,N)j =1

2π× 1

B`,N×√`

N.

` = 4, N = 3, X`,N =1

256

B =√

1− 64X , B4,3 =√

1− 64/256 =

√3

2, by Hypothesis 3.

1

2π× 1

B`,N×√`

N=

1

2π× 2√

3×√

4

3=

2

3× 1

π.

∞∑j=0

(2j

j

)3(j +

1

6

)(1

256

)j

=2

3× 1

π.

Page 57: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

∞∑j=0

h(j) (j + λ`,N) (X`,N)j =1

2π× 1

B`,N×√`

N.

` = 4, N = 3, X`,N =1

256

B =√

1− 64X , B4,3 =√

1− 64/256 =

√3

2, by Hypothesis 3.

1

2π× 1

B`,N×√`

N=

1

2π× 2√

3×√

4

3=

2

3× 1

π.

∞∑j=0

(2j

j

)3(j +

1

6

)(1

256

)j

=2

3× 1

π.

Page 58: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

∞∑j=0

h(j) (j + λ`,N) (X`,N)j =1

2π× 1

B`,N×√`

N.

` = 4, N = 3, X`,N =1

256

B =√

1− 64X ,

B4,3 =√

1− 64/256 =

√3

2, by Hypothesis 3.

1

2π× 1

B`,N×√`

N=

1

2π× 2√

3×√

4

3=

2

3× 1

π.

∞∑j=0

(2j

j

)3(j +

1

6

)(1

256

)j

=2

3× 1

π.

Page 59: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

∞∑j=0

h(j) (j + λ`,N) (X`,N)j =1

2π× 1

B`,N×√`

N.

` = 4, N = 3, X`,N =1

256

B =√

1− 64X , B4,3 =√

1− 64/256 =

√3

2, by Hypothesis 3.

1

2π× 1

B`,N×√`

N=

1

2π× 2√

3×√

4

3=

2

3× 1

π.

∞∑j=0

(2j

j

)3(j +

1

6

)(1

256

)j

=2

3× 1

π.

Page 60: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

∞∑j=0

h(j) (j + λ`,N) (X`,N)j =1

2π× 1

B`,N×√`

N.

` = 4, N = 3, X`,N =1

256

B =√

1− 64X , B4,3 =√

1− 64/256 =

√3

2, by Hypothesis 3.

1

2π× 1

B`,N×√`

N=

1

2π× 2√

3×√

4

3=

2

3× 1

π.

∞∑j=0

(2j

j

)3(j +

1

6

)(1

256

)j

=2

3× 1

π.

Page 61: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

∞∑j=0

h(j) (j + λ`,N) (X`,N)j =1

2π× 1

B`,N×√`

N.

` = 4, N = 3, X`,N =1

256

B =√

1− 64X , B4,3 =√

1− 64/256 =

√3

2, by Hypothesis 3.

1

2π× 1

B`,N×√`

N=

1

2π× 2√

3×√

4

3=

2

3× 1

π.

∞∑j=0

(2j

j

)3(j +

1

6

)(1

256

)j

=2

3× 1

π.

Page 62: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

λ =X

2

d

dX

(Y B(Y )

X B(X )÷ dY

dX

),

X 4+Y 4−16777216X 3Y 3+294912 (X 3Y 2+X 2Y 3)−900(X 3Y+XY 3)

+28422X 2Y 2 + 72(X 2Y + XY 2)− XY = 0,

B(X ) =√

1− 64X .

Then, put X = Y = 1/256 to get λ = 1/6.

∞∑j=0

(2j

j

)3(j +

1

6

)(1

256

)j

=2

3× 1

π.

This is Ramanujan’s formula (28).

Page 63: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

λ =X

2

d

dX

(Y B(Y )

X B(X )÷ dY

dX

),

X 4+Y 4−16777216X 3Y 3+294912 (X 3Y 2+X 2Y 3)−900(X 3Y+XY 3)

+28422X 2Y 2 + 72(X 2Y + XY 2)− XY = 0,

B(X ) =√

1− 64X .

Then, put X = Y = 1/256 to get λ = 1/6.

∞∑j=0

(2j

j

)3(j +

1

6

)(1

256

)j

=2

3× 1

π.

This is Ramanujan’s formula (28).

Page 64: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

λ =X

2

d

dX

(Y B(Y )

X B(X )÷ dY

dX

),

X 4+Y 4−16777216X 3Y 3+294912 (X 3Y 2+X 2Y 3)−900(X 3Y+XY 3)

+28422X 2Y 2 + 72(X 2Y + XY 2)− XY = 0,

B(X ) =√

1− 64X .

Then, put X = Y = 1/256 to get λ = 1/6.

∞∑j=0

(2j

j

)3(j +

1

6

)(1

256

)j

=2

3× 1

π.

This is Ramanujan’s formula (28).

Page 65: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

λ =X

2

d

dX

(Y B(Y )

X B(X )÷ dY

dX

),

X 4+Y 4−16777216X 3Y 3+294912 (X 3Y 2+X 2Y 3)−900(X 3Y+XY 3)

+28422X 2Y 2 + 72(X 2Y + XY 2)− XY = 0,

B(X ) =√

1− 64X .

Then, put X = Y = 1/256 to get λ = 1/6.

∞∑j=0

(2j

j

)3(j +

1

6

)(1

256

)j

=2

3× 1

π.

This is Ramanujan’s formula (28).

Page 66: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Other examples

∞∑j=0

(2j

j

)3 (42j + 5)

212j=

16

π

Ramanujan–Disney series

Page 67: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Other examples

∞∑j=0

(2j

j

)3 (42j + 5)

212j=

16

π

Ramanujan–Disney series

Page 68: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Other examples

∞∑j=0

(2j

j

)3 (42j + 5)

212j=

16

π

Ramanujan–Disney series

Page 69: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Other examples:

∞∑n=0

(4n

2n

)(2n

n

)2(n +

1103

26390

)(1

396

)4n

=9801

√2

105560× 1

π.

It is Ramanujan’s formula (44), his last example.

Each term adds 8 decimal digits.

It comes from the level 2 theory, with degree 29.

Used by W. Gosper in 1985 to compute over 17,500,000 decimaldigits of π.

The value 1103 hadn’t been proved when Gosper set the record.

There are now three proofs of this result, all within the last twoyears.

Page 70: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Other examples:

∞∑n=0

(4n

2n

)(2n

n

)2(n +

1103

26390

)(1

396

)4n

=9801

√2

105560× 1

π.

It is Ramanujan’s formula (44), his last example.

Each term adds 8 decimal digits.

It comes from the level 2 theory, with degree 29.

Used by W. Gosper in 1985 to compute over 17,500,000 decimaldigits of π.

The value 1103 hadn’t been proved when Gosper set the record.

There are now three proofs of this result, all within the last twoyears.

Page 71: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Other examples:

∞∑n=0

(4n

2n

)(2n

n

)2(n +

1103

26390

)(1

396

)4n

=9801

√2

105560× 1

π.

It is Ramanujan’s formula (44), his last example.

Each term adds 8 decimal digits.

It comes from the level 2 theory, with degree 29.

Used by W. Gosper in 1985 to compute over 17,500,000 decimaldigits of π.

The value 1103 hadn’t been proved when Gosper set the record.

There are now three proofs of this result, all within the last twoyears.

Page 72: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Other examples:

∞∑n=0

(4n

2n

)(2n

n

)2(n +

1103

26390

)(1

396

)4n

=9801

√2

105560× 1

π.

It is Ramanujan’s formula (44), his last example.

Each term adds 8 decimal digits.

It comes from the level 2 theory, with degree 29.

Used by W. Gosper in 1985 to compute over 17,500,000 decimaldigits of π.

The value 1103 hadn’t been proved when Gosper set the record.

There are now three proofs of this result, all within the last twoyears.

Page 73: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Other examples:

∞∑n=0

(4n

2n

)(2n

n

)2(n +

1103

26390

)(1

396

)4n

=9801

√2

105560× 1

π.

It is Ramanujan’s formula (44), his last example.

Each term adds 8 decimal digits.

It comes from the level 2 theory, with degree 29.

Used by W. Gosper in 1985 to compute over 17,500,000 decimaldigits of π.

The value 1103 hadn’t been proved when Gosper set the record.

There are now three proofs of this result, all within the last twoyears.

Page 74: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Other examples:

∞∑n=0

(4n

2n

)(2n

n

)2(n +

1103

26390

)(1

396

)4n

=9801

√2

105560× 1

π.

It is Ramanujan’s formula (44), his last example.

Each term adds 8 decimal digits.

It comes from the level 2 theory, with degree 29.

Used by W. Gosper in 1985 to compute over 17,500,000 decimaldigits of π.

The value 1103 hadn’t been proved when Gosper set the record.

There are now three proofs of this result, all within the last twoyears.

Page 75: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Other examples:

∞∑n=0

(4n

2n

)(2n

n

)2(n +

1103

26390

)(1

396

)4n

=9801

√2

105560× 1

π.

It is Ramanujan’s formula (44), his last example.

Each term adds 8 decimal digits.

It comes from the level 2 theory, with degree 29.

Used by W. Gosper in 1985 to compute over 17,500,000 decimaldigits of π.

The value 1103 hadn’t been proved when Gosper set the record.

There are now three proofs of this result, all within the last twoyears.

Page 76: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

∞∑n=0

(6n

3n

)(3n

n

)(2n

n

)(n +

13591409

545140134

)(−1

640320

)3n

=26 × 53/2 × 233/2 × 293/2

33/2 × 7× 11× 19× 127× 163× 1

π

David and Gregory Chudnovsky (1987).

Level 1, degree 163.

Each term contributes roughly 14 decimal digits per term.

Used by the Chudnovskys (and others) in world-record calculationsof π.

Page 77: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

∞∑n=0

(6n

3n

)(3n

n

)(2n

n

)(n +

13591409

545140134

)(−1

640320

)3n

=26 × 53/2 × 233/2 × 293/2

33/2 × 7× 11× 19× 127× 163× 1

π

David and Gregory Chudnovsky (1987).

Level 1, degree 163.

Each term contributes roughly 14 decimal digits per term.

Used by the Chudnovskys (and others) in world-record calculationsof π.

Page 78: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

∞∑n=0

(6n

3n

)(3n

n

)(2n

n

)(n +

13591409

545140134

)(−1

640320

)3n

=26 × 53/2 × 233/2 × 293/2

33/2 × 7× 11× 19× 127× 163× 1

π

David and Gregory Chudnovsky (1987).

Level 1, degree 163.

Each term contributes roughly 14 decimal digits per term.

Used by the Chudnovskys (and others) in world-record calculationsof π.

Page 79: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

∞∑n=0

(6n

3n

)(3n

n

)(2n

n

)(n +

13591409

545140134

)(−1

640320

)3n

=26 × 53/2 × 233/2 × 293/2

33/2 × 7× 11× 19× 127× 163× 1

π

David and Gregory Chudnovsky (1987).

Level 1, degree 163.

Each term contributes roughly 14 decimal digits per term.

Used by the Chudnovskys (and others) in world-record calculationsof π.

Page 80: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Data for h(j) =(

6j3j

)(3jj

)(2jj

)q, ` N x λ Ramanujan

2 1203

328

q = e−2π√N 3 4

603111 (33)

` = 1 4 1663

563

7 12553

8133 (34)

7 −1153

863

11 −1323

15154

q = −e−π√N 19 −1

96325342

` = 4 27 −94803

31506

43 −19603

2635418

67 −152803

10177261702

163 −16403203

13591409545140134

Page 81: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Data for h(j) =(

4j2j

)(2jj

)2

q, ` N x λ Ramanujan

2 264

17

3 1482

18 (40)

q = e−2π√

N/2 5 1124

110 (41)

` = 2 9 1284

340 (42)

11 115842

19280 (43)

29 13964

110326390 (44)

5 −1210

320 (35)

7 −1632

865

q = −e−π√N 9 −3

1922328 (36)

` = 4 13 −13242

23260 (37)

25 −53602

41644 (38)

37 −16722

112321460 (39)

Page 82: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Data for h(j) =(

3jj

)(2jj

)2

q, ` N x λ Ramanujan

2 163

16

e−2π√

N/3 4 4183

215 (31)

5 1153

433 (32)

9 −1192

15

17 −1123

751

−e−π√

N/3 25 −25603

19

41 −1483

53615

49 −492523

13165

89 −13003

82714151

Page 83: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Data for h(j) =(

2jj

)3

q = e−π√N 3 1

25616 (28)

` = 4 7 14096

542 (29)

q = −e−π√N 2 −1

6414

` = 4 4 −1512

16

Page 84: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Directions for further research

Page 85: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Sporadic sequences

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) 6∈ Q

Not obvious from the recurrence relation that tj is always aninteger.

Page 86: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Sporadic sequences

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) 6∈ Q

Not obvious from the recurrence relation that tj is always aninteger.

Page 87: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Sporadic sequences

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) 6∈ Q

Not obvious from the recurrence relation that tj is always aninteger.

Page 88: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Apery, 1978

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

sk =k∑

j=0

(k

j

)2(k + j

j

)

Franel, 1894

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

sk =k∑

j=0

(k

j

)3

Zagier, 1998, 2009

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

Page 89: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Apery, 1978

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

sk =k∑

j=0

(k

j

)2(k + j

j

)

Franel, 1894

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

sk =k∑

j=0

(k

j

)3

Zagier, 1998, 2009

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

Page 90: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

(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(2j

j

)(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

)

Page 91: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Analogue 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

Page 92: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Analogue 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

Page 93: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

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

(a, b, c) Level

(11, 3, 1) 5

(−17,−6,−72) 6

(10, 3,−9) 6

(7, 2, 8) 6

(12, 4,−32) 8

(−9,−3,−27) 9

The sequences satisfy several congruence properties.

Page 94: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Another computer search (C., 2012)

(n + 1)3vn+1 = (2n + 1)(an2 + an + b)vn + n(cn2 + d)vn−1

where v0 = 1, v−1 = 0.

(a, b, c , d) = (13, 4, 27,−3): appears to be integer valued.

What are the parameterizing modular forms?

z =∞∑n=0

vn xn, z =?, x =?

z =7P(q7)− P(q)

6, x =

x

1 + 13y + 49y2,

where

y := q∞∏j=1

(1− q7j)4

(1− qj)4

Page 95: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Another computer search (C., 2012)

(n + 1)3vn+1 = (2n + 1)(an2 + an + b)vn + n(cn2 + d)vn−1

where v0 = 1, v−1 = 0.

(a, b, c , d) = (13, 4, 27,−3): appears to be integer valued.

What are the parameterizing modular forms?

z =∞∑n=0

vn xn, z =?, x =?

z =7P(q7)− P(q)

6, x =

x

1 + 13y + 49y2,

where

y := q∞∏j=1

(1− q7j)4

(1− qj)4

Page 96: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Another computer search (C., 2012)

(n + 1)3vn+1 = (2n + 1)(an2 + an + b)vn + n(cn2 + d)vn−1

where v0 = 1, v−1 = 0.

(a, b, c , d) = (13, 4, 27,−3): appears to be integer valued.

What are the parameterizing modular forms?

z =∞∑n=0

vn xn, z =?, x =?

z =7P(q7)− P(q)

6, x =

x

1 + 13y + 49y2,

where

y := q∞∏j=1

(1− q7j)4

(1− qj)4

Page 97: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Another computer search (C., 2012)

(n + 1)3vn+1 = (2n + 1)(an2 + an + b)vn + n(cn2 + d)vn−1

where v0 = 1, v−1 = 0.

(a, b, c , d) = (13, 4, 27,−3): appears to be integer valued.

What are the parameterizing modular forms?

z =∞∑n=0

vn xn, z =?, x =?

z =7P(q7)− P(q)

6, x =

x

1 + 13y + 49y2,

where

y := q∞∏j=1

(1− q7j)4

(1− qj)4

Page 98: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Another computer search (C., 2012)

(n + 1)3vn+1 = (2n + 1)(an2 + an + b)vn + n(cn2 + d)vn−1

where v0 = 1, v−1 = 0.

(a, b, c , d) = (13, 4, 27,−3): appears to be integer valued.

What are the parameterizing modular forms?

z =∞∑n=0

vn xn, z =?, x =?

z =7P(q7)− P(q)

6, x =

x

1 + 13y + 49y2,

where

y := q∞∏j=1

(1− q7j)4

(1− qj)4

Page 99: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

(n + 1)3vn+1 = (2n + 1)(an2 + an + b)vn + n(cn2 + d)vn−1

where v0 = 1, v−1 = 0.

(a, b, c , d) = (13, 4, 27,−3): appears to be integer valued.

In fact, W. Zudilin found that

vn =n∑

j=0

(n

j

)2(2j

n

)(n + j

j

).

Series for 1/π can be obtained, as before, e.g.,

1

π=√

7∞∑n=0

(−1)nvn(11895n + 1286)

223n+3.

Level 7, degree 61.

Other sporadic sequences, e.g., complex? Four term relations?

Page 100: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

(n + 1)3vn+1 = (2n + 1)(an2 + an + b)vn + n(cn2 + d)vn−1

where v0 = 1, v−1 = 0.

(a, b, c , d) = (13, 4, 27,−3): appears to be integer valued.

In fact, W. Zudilin found that

vn =n∑

j=0

(n

j

)2(2j

n

)(n + j

j

).

Series for 1/π can be obtained, as before, e.g.,

1

π=√

7∞∑n=0

(−1)nvn(11895n + 1286)

223n+3.

Level 7, degree 61.

Other sporadic sequences, e.g., complex? Four term relations?

Page 101: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

(n + 1)3vn+1 = (2n + 1)(an2 + an + b)vn + n(cn2 + d)vn−1

where v0 = 1, v−1 = 0.

(a, b, c , d) = (13, 4, 27,−3): appears to be integer valued.

In fact, W. Zudilin found that

vn =n∑

j=0

(n

j

)2(2j

n

)(n + j

j

).

Series for 1/π can be obtained, as before, e.g.,

1

π=√

7∞∑n=0

(−1)nvn(11895n + 1286)

223n+3.

Level 7, degree 61.

Other sporadic sequences, e.g., complex? Four term relations?

Page 102: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

2. Clausen-type formulas

Chan, Tanigawa, Yang and Zudilin (2011)

(1 + cw2)

( ∞∑k=0

skwk

)2

=∞∑k=0

(2k

k

)sk

(w(1− aw − cw2)

(1 + cw2)2

)k

Almkvist, van Straten and Zudilin (2011)

(1− aw − cw2)

( ∞∑k=0

skwk

)2

=∞∑k=0

tk

(w

1− aw − cw2

)k

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

(k + 1)3tk+1 = −(2k + 1)(ak2 + ak + a− 2b)tk − (4c + a2)k3tk−1

s0 = t0 = 1 s−1 = t−1 = 0

Page 103: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

2. Clausen-type formulas

Chan, Tanigawa, Yang and Zudilin (2011)

(1 + cw2)

( ∞∑k=0

skwk

)2

=∞∑k=0

(2k

k

)sk

(w(1− aw − cw2)

(1 + cw2)2

)k

Almkvist, van Straten and Zudilin (2011)

(1− aw − cw2)

( ∞∑k=0

skwk

)2

=∞∑k=0

tk

(w

1− aw − cw2

)k

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

(k + 1)3tk+1 = −(2k + 1)(ak2 + ak + a− 2b)tk − (4c + a2)k3tk−1

s0 = t0 = 1 s−1 = t−1 = 0

Page 104: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

2. Clausen-type formulas

Chan, Tanigawa, Yang and Zudilin (2011)

(1 + cw2)

( ∞∑k=0

skwk

)2

=∞∑k=0

(2k

k

)sk

(w(1− aw − cw2)

(1 + cw2)2

)k

Almkvist, van Straten and Zudilin (2011)

(1− aw − cw2)

( ∞∑k=0

skwk

)2

=∞∑k=0

tk

(w

1− aw − cw2

)k

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

(k + 1)3tk+1 = −(2k + 1)(ak2 + ak + a− 2b)tk − (4c + a2)k3tk−1

s0 = t0 = 1 s−1 = t−1 = 0

Page 105: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

2. Clausen-type formulas

Chan, Tanigawa, Yang and Zudilin (2011)

(1 + cw2)

( ∞∑k=0

skwk

)2

=∞∑k=0

(2k

k

)sk

(w(1− aw − cw2)

(1 + cw2)2

)k

Almkvist, van Straten and Zudilin (2011)

(1− aw − cw2)

( ∞∑k=0

skwk

)2

=∞∑k=0

tk

(w

1− aw − cw2

)k

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

(k + 1)3tk+1 = −(2k + 1)(ak2 + ak + a− 2b)tk − (4c + a2)k3tk−1

s0 = t0 = 1 s−1 = t−1 = 0

Page 106: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

3. q-analogues

(x + y)n =n∑

j=0

(n

j

)xn−jyk

(n

j

)=

n!

j!(n − j)!

Exercise: find and prove a formula for the coefficients c(n, j , q) inthe expansion of

(x + y)(x + qy)(x + q2y) · · · (x + qn−1y) =n∑

j=0

c(n, j , q)xn−jy j .

Called: the q-binomial theorem. (See: Polya and Alexanderson)

Page 107: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

3. q-analogues

(x + y)n =n∑

j=0

(n

j

)xn−jyk

(n

j

)=

n!

j!(n − j)!

Exercise: find and prove a formula for the coefficients c(n, j , q) inthe expansion of

(x + y)(x + qy)(x + q2y) · · · (x + qn−1y) =n∑

j=0

c(n, j , q)xn−jy j .

Called: the q-binomial theorem. (See: Polya and Alexanderson)

Page 108: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

3. q-analogues

(x + y)n =n∑

j=0

(n

j

)xn−jyk

(n

j

)=

n!

j!(n − j)!

Exercise: find and prove a formula for the coefficients c(n, j , q) inthe expansion of

(x + y)(x + qy)(x + q2y) · · · (x + qn−1y) =n∑

j=0

c(n, j , q)xn−jy j .

Called: the q-binomial theorem. (See: Polya and Alexanderson)

Page 109: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

3. q-analogues

(x + y)n =n∑

j=0

(n

j

)xn−jyk

(n

j

)=

n!

j!(n − j)!

Exercise: find and prove a formula for the coefficients c(n, j , q) inthe expansion of

(x + y)(x + qy)(x + q2y) · · · (x + qn−1y) =n∑

j=0

c(n, j , q)xn−jy j .

Called: the q-binomial theorem. (See: Polya and Alexanderson)

Page 110: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

q-analogues

n!q = 1× (1 + q)× (1 + q + q2)× · · · × (1 + q + q2 + · · · qn−1)

=(1− q)

(1− q)× (1− q2)

(1− q)× (1− q3)

(1− q)× · · · × (1− qn)

(1− q)

→ 1× 2× 3× · · · × n = n! as q → 1.

Define the q-integer [n]q by

[n]q = 1 + q + q2 + · · ·+ qn−1 =1− qn

1− q.

Then n!q = [1]q × [2]q × · · · × [n]q.

Page 111: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

q-analogues

n!q = 1× (1 + q)× (1 + q + q2)× · · · × (1 + q + q2 + · · · qn−1)

=(1− q)

(1− q)× (1− q2)

(1− q)× (1− q3)

(1− q)× · · · × (1− qn)

(1− q)

→ 1× 2× 3× · · · × n = n! as q → 1.

Define the q-integer [n]q by

[n]q = 1 + q + q2 + · · ·+ qn−1 =1− qn

1− q.

Then n!q = [1]q × [2]q × · · · × [n]q.

Page 112: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

q-analogues

n!q = 1× (1 + q)× (1 + q + q2)× · · · × (1 + q + q2 + · · · qn−1)

=(1− q)

(1− q)× (1− q2)

(1− q)× (1− q3)

(1− q)× · · · × (1− qn)

(1− q)

→ 1× 2× 3× · · · × n = n! as q → 1.

Define the q-integer [n]q by

[n]q = 1 + q + q2 + · · ·+ qn−1 =1− qn

1− q.

Then n!q = [1]q × [2]q × · · · × [n]q.

Page 113: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

q-analogues

n!q = 1× (1 + q)× (1 + q + q2)× · · · × (1 + q + q2 + · · · qn−1)

=(1− q)

(1− q)× (1− q2)

(1− q)× (1− q3)

(1− q)× · · · × (1− qn)

(1− q)

→ 1× 2× 3× · · · × n = n! as q → 1.

Define the q-integer [n]q by

[n]q = 1 + q + q2 + · · ·+ qn−1 =1− qn

1− q.

Then n!q = [1]q × [2]q × · · · × [n]q.

Page 114: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

q-analogues

n!q = 1× (1 + q)× (1 + q + q2)× · · · × (1 + q + q2 + · · · qn−1)

=(1− q)

(1− q)× (1− q2)

(1− q)× (1− q3)

(1− q)× · · · × (1− qn)

(1− q)

→ 1× 2× 3× · · · × n = n! as q → 1.

Define the q-integer [n]q by

[n]q = 1 + q + q2 + · · ·+ qn−1 =1− qn

1− q.

Then n!q = [1]q × [2]q × · · · × [n]q.

Page 115: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Ramanujan’s formula

∞∑j=0

(2j

j

)3(j +

1

6

)(1

256

)j

=2

3× 1

π

has the q-analogue

∞∑j=0

qj2[6j + 1]q

(q; q2)2k(q2; q4)k(q4; q4)3k

=(1 + q)(q2; q4)∞(q6; q4)∞

(q4; q4)2∞.

Notation:

(x ; q)∞ = (1− x)(1− qx)(1− q2x)(1− q3x) · · · , |q| < 1.

Victor J. W. Guo and Ji-Cai Liu.

Page 116: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Ramanujan’s formula

∞∑j=0

(2j

j

)3(j +

1

6

)(1

256

)j

=2

3× 1

π

has the q-analogue

∞∑j=0

qj2[6j + 1]q

(q; q2)2k(q2; q4)k(q4; q4)3k

=(1 + q)(q2; q4)∞(q6; q4)∞

(q4; q4)2∞.

Notation:

(x ; q)∞ = (1− x)(1− qx)(1− q2x)(1− q3x) · · · , |q| < 1.

Victor J. W. Guo and Ji-Cai Liu.

Page 117: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Ramanujan’s formula

∞∑j=0

(2j

j

)3(j +

1

6

)(1

256

)j

=2

3× 1

π

has the q-analogue

∞∑j=0

qj2[6j + 1]q

(q; q2)2k(q2; q4)k(q4; q4)3k

=(1 + q)(q2; q4)∞(q6; q4)∞

(q4; q4)2∞.

Notation:

(x ; q)∞ = (1− x)(1− qx)(1− q2x)(1− q3x) · · · , |q| < 1.

Victor J. W. Guo and Ji-Cai Liu.

Page 118: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

Ramanujan’s formula

∞∑j=0

(2j

j

)3(j +

1

6

)(1

256

)j

=2

3× 1

π

has the q-analogue

∞∑j=0

qj2[6j + 1]q

(q; q2)2k(q2; q4)k(q4; q4)3k

=(1 + q)(q2; q4)∞(q6; q4)∞

(q4; q4)2∞.

Notation:

(x ; q)∞ = (1− x)(1− qx)(1− q2x)(1− q3x) · · · , |q| < 1.

Victor J. W. Guo and Ji-Cai Liu.

Page 119: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

The last slide.

1. Elliptic functions

2. Modular functions

3. Jacobi’s inversion formula

4. Parametrization of hypergeometric and Heun functions bymodular forms

5. All of the above lead to Ramanujan-type series for 1/π.

6. Many ideas are involved. That leads to fruitful researchquestions.

The end.

Page 120: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

The last slide.

1. Elliptic functions

2. Modular functions

3. Jacobi’s inversion formula

4. Parametrization of hypergeometric and Heun functions bymodular forms

5. All of the above lead to Ramanujan-type series for 1/π.

6. Many ideas are involved. That leads to fruitful researchquestions.

The end.

Page 121: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

The last slide.

1. Elliptic functions

2. Modular functions

3. Jacobi’s inversion formula

4. Parametrization of hypergeometric and Heun functions bymodular forms

5. All of the above lead to Ramanujan-type series for 1/π.

6. Many ideas are involved. That leads to fruitful researchquestions.

The end.

Page 122: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

The last slide.

1. Elliptic functions

2. Modular functions

3. Jacobi’s inversion formula

4. Parametrization of hypergeometric and Heun functions bymodular forms

5. All of the above lead to Ramanujan-type series for 1/π.

6. Many ideas are involved. That leads to fruitful researchquestions.

The end.

Page 123: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

The last slide.

1. Elliptic functions

2. Modular functions

3. Jacobi’s inversion formula

4. Parametrization of hypergeometric and Heun functions bymodular forms

5. All of the above lead to Ramanujan-type series for 1/π.

6. Many ideas are involved. That leads to fruitful researchquestions.

The end.

Page 124: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

The last slide.

1. Elliptic functions

2. Modular functions

3. Jacobi’s inversion formula

4. Parametrization of hypergeometric and Heun functions bymodular forms

5. All of the above lead to Ramanujan-type series for 1/π.

6. Many ideas are involved. That leads to fruitful researchquestions.

The end.

Page 125: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

The last slide.

1. Elliptic functions

2. Modular functions

3. Jacobi’s inversion formula

4. Parametrization of hypergeometric and Heun functions bymodular forms

5. All of the above lead to Ramanujan-type series for 1/π.

6. Many ideas are involved. That leads to fruitful researchquestions.

The end.

Page 126: Ramanujan’s series for 1=ˇ Cooper3.pdf · Srinivasa Ramanujan, 1887 { 1920 Jeremy Irons and Dev Patel as G. H. Hardy and S. Ramanujan in \The Man who knew In nity", based on a

The last slide.

1. Elliptic functions

2. Modular functions

3. Jacobi’s inversion formula

4. Parametrization of hypergeometric and Heun functions bymodular forms

5. All of the above lead to Ramanujan-type series for 1/π.

6. Many ideas are involved. That leads to fruitful researchquestions.

The end.