Ramanujan s theories of elliptic functions to alternative bases, and beyond.

Transcription

1 Ramanuan s theories of elliptic functions to alternative bases, and beyond. Shaun Cooper Massey University, Auckland Askey 80 Conference. December 6, 03.

2 Outline Sporadic sequences Background: classical theory 3 Ramanuan s alternative theories 4 Beyond Ramanuan s alternative theories

3 Roger Apéry, Classical results n= n = π 6, Apéry s theorem (978) n= n 4 = π4 90, n= n 6 = π6 945, ζ(3) := n= n 3 is irrational Apéry introduced two sequences of integers: one to prove ζ() Q (well-known) and the other to prove ζ(3) Q (new).

4 Apéry s numbers used to prove ζ() Q (k + ) s k+ = (k + k + 3)s k + k s k, s 0 = The numbers s k are integers not obvious. Apéry s numbers used to prove ζ(3) Q (k + ) 3 t k+ =(k + )(7k + 7k + 5)t k k 3 t k, t 0 = The numbers t k are integers not obvious.

5 Apéry s numbers used to prove ζ() Q (k + ) s k+ = (k + k + 3)s k + k s k, s 0 = s k = k k k + Proof: by the method of creative telescoping. Now, automated. The numbers s k are integers now obvious from the binomial sum.

6 F. Beukers, 983 (k + ) s k+ = (k + k + 3)s k + k s k, s 0 = z = s k x k k=0 x = q z = = = ( q 5 4 ) 5 ( q 5 ) 5 ( q 5 3 ) 5 ( q 5 ) 5 = r 5(q) 5 ( q ) ( q 5 4 ) 5 ( q 5 ) 5. Another proof that s k is an integer. r 5 (q): the Rogers-Ramanuan continued fraction 3. How to find other examples?

7 Sporadic sequences Franel, 894 (k + ) s k+ =(7k +7k + )s k +8k s k, s 0 = s k = k k 3 Apéry, 978 (k + ) s k+ = (k + k + 3)s k + k s k, s 0 = k k k + s k = Zagier, 998, 009 (k + ) s k+ =(ak + ak + b)s k + ck s k, s 0 =

8 (k +) s k+ =(ak + ak + b)s k + ck s k, s 0 = (a, b, c) (, 3, ) ( 7, 6, 7) (0, 3, 9) (7,, 8) (, 4, 3) ( 9, 3, 7) s(k) k k + k 3 ( 8) k, k k 3 k 4 k k k k ( 3) k 3

9 Analogues of Beukers result (k + ) s k+ =(7k +7k + )s k +8k s k, s 0 = z = s k x k = k=0 k k=0 k 3 x k x = q = z = ( q ) 3 ( q 6 ) 9 ( q ) 3 ( q 3 ) 9 = r c(q) 3 = ( q )( q 3 ) 6 ( q ) ( q 6 ) 3. r c (q) is Ramanuan s cubic continued fraction. Similar results hold for Zagier s other examples

10 Apéry s numbers used to prove ζ() Q (k + ) s k+ = (k + k + 3)s k + k s k, s 0 =. Three-term recurrence relation. Coefficients are polynomials of degree 3. Zagier s sporadic sequences (k + ) s k+ =(ak + ak + b)s k + ck s k, s 0 = Apéry s numbers used to prove ζ(3) Q (k + ) 3 t k+ =(k + )(7k + 7k + 5)t k k 3 t k, t 0 =. Three-term recurrence relation. Coefficients are polynomials of degree 3

11 T. Sato, Abstract for Math. Soc. Japan, March 00 (k + ) 3 t k+ =(k + )(7k + 7k + 5)t k k 3 t k, t 0 = x = q = ( q ) ( q 6 ) ( q ) ( q 3 ) (k + ) 3 t k+ = (k + )(k + k + 5)t k 5k 3 t k, t 0 = Analogues of π = 980 k=0 k k 4k ( k) k 396 4k. H. H. Chan and coauthors, series of papers

12 Apéry numbers and modular forms f (5) := 5P(q5 ) P(q) 4 = k k=0 + k k η η5 f (5) f (6) := 30P(q6 ) 3P(q 3 )+P(q ) 5P(q) 4 + k η η η 3 η 6 = k k f (6) k=0 η n = q n/4 ( q n q ), P(q) = 4 q = =

13 Outline Sporadic sequences Background: classical theory 3 Ramanuan s alternative theories 4 Beyond Ramanuan s alternative theories

14 Jacobi, 89 x = n= n= q (n+ ) q n 4, q < x(0) = 0 x() = dx > 0 dq dx dq = 6 q=0 Figure: Graph of x versus q

15 Hypergeometric function (a) n = a(a + )(a + ) (a + n ), n Z + ; (a) 0 = F (a, b; c; x) = n=0 (a) n (b) n (c) n n! x n 3F (a, b, c; d, e; x) = n=0 (a) n (b) n (c) n (d) n (e) n n! x n 0F 0 ( ; ; x) = F 0 (a; ; x) = n=0 n=0 n! x n (a) n n! x n

16 Jacobi s inversion formula, 89 x = n= n= q (n+ ) q n 4, q =exp π F (, ; ; x) F (, ; ; x) x(0) = 0 x() = x(e π ) = / x(e πt )+x(e π/t ) =, t > 0 Figure: Graph of x versus q

17 Jacobi s inversion formula, 89 x = n= n= q (n+ ) q n 4, q =exp π F (, ; ; x) F (, ; ; x) Figure: Graph of x versus q z = F (, ; ; x) = q dx dq = z x( x) n= q n

18 Squares of hypergeometric functions 0F 0 ( ; ; x) =(e x ) = 0 F 0 ( ; ;x) F 0 (a; ; x) = ( x) a = F 0 (a; ; x) F (a, b; c; x) =

19 Clausen s identity (88) Clausen s identity F +, ; ; x = 3 F +,, ;, ; 4x( x) Clausen + Jacobi n= q n 4 = 3 F,, ;, ; 4x( x) After some manipulations (this expression generalizes) f (4) := 4P(q4 ) P(q) 3 η 4 = η4 4 3 η 4 f (4)

20 Aside: Jacobi s sum of four squares theorem n= q n 4 =+8 = 4 q q # x + x + x 3 + x 4 = n, x, x, x 3, x 4 Z =8 d n 4d d Corollary (Lagrange) Every positive integer is a sum of four squares. # x + x + x 3 + x 4 = n > 0

21 Another aside: Ramanuan (94) π = 6 3 (4 + 5)

22 Outline Sporadic sequences Background: classical theory 3 Ramanuan s alternative theories 4 Beyond Ramanuan s alternative theories

23 Ramanuan (94) There are similar theories when q =exp π is replaced by any of z( x) z(x), z(x) = F, ; ; x q =exp π z ( x), z (x) = F z (x) 4, 3 4 ; ; x q =exp π z ( x) 3 z (x), z (x) = F 3, 3 ; ; x q 3 =exp π z 3( x), z 3 (x) = F z 3 (x) 6, 5 6 ; ; x

24 Ramanuan s alternative theories of elliptic functions Ramanuan (94) q =exp π F 3, 3 ; ; x 3 F 3, 3 ; ; x J. M. Borwein and P. B. Borwein (99) x = m= n= q (m+ 3 ) +(m+ 3 )(n+ 3 )+(n+ 3 ) m= n= q m +mn+n 3

25 Ramanuan s alternative theories of elliptic functions q =exp π F 3, 3 ; ; x 3 F 3, 3 ; ; x x(0) = 0 x() = x(e π/ 3 ) = / Figure: Graph of x versus q

26 Ramanuan s alternative theories of elliptic functions q =exp π F 3, 3 ; ; x 3 F 3, 3 ; ; x z = F 3, 3 ; ; x q dx dq = m= n= = z x( x) q m +mn+n Figure: Graph of x versus q

27 Ramanuan s alternative theories of elliptic functions Ramanuan pp. 57 6, second notebook 7 Feb 93, second letter to G. H. Hardy 94 paper Modular equations and approximations to /π 7 series for /π Mordell (97), Watson (93) It is unfortunate that Ramanuan has not developed in detail the corresponding theories... There are developments of functions analogous to elliptic functions which I have not seen elsewhere... Fricke (96) Inversion formula for F ( 6, 5 6 ; ; x)

28 Ramanuan s alternative theories of elliptic functions K. Venkatachaliengar (988, republished 0) Initial investigations into the alternative theories J. M. Borwein and P. B. Borwein ( ) A book and a series of papers Proved all 7 of Ramanuan s series for /π Discovered the cubic theta function q m +mn+n Berndt, Bhargava and Garvan (995) Proved all of the results on pp of Ramanuan s second notebook. (Trans. Amer. Math. Soc., 8 pages)

29 Ramanuan s alternative theories of elliptic functions H. H. Chan (998) The F ( 3, 3 ; ; x) theory Berndt, Chan and Liaw (00) The F ( 4, 3 4 ; ; x) theory K. S. Williams (004) The F ( 3, 3 ; ; x) theory C., (009) A unified treatment for all four theories D. Schultz (03) Cubic theory

30 Example of a modular form Eisenstein series E n (τ) = Transformations (,k)=(0,0), n =, 3,..., Im(τ) > 0 ( + kτ) n E n (τ + ) = E n (τ) E n = τ n E n (τ) τ E n is a modular form of weight n aτ + b E n =(cτ + d) n E n (τ) cτ + d for all integers a, b, c and d with ad bc =.

31 The effect of scaling Suppose f aτ + b =(cτ + d) n f (τ) cτ + d for all integers a, b, c and d with ad bc =. Let m be a positive integer and let g(τ) =f (mτ). Then aτ + b g =(cτ + d) n g(τ) cτ + d for all integers a, b, c and d with ad bc =, provided in addition c 0(modm).

32 Congruence subgroups The modular group Γ = a c b d a, b, c, d Z, ad bc = Congruence subgroup Γ 0 (m) = a c b d Γ c 0(modm) Modular form A function f is a modular form of weight n and level m if aτ + b f =(cτ + d) n a b f (τ) for all Γ cτ + d c d 0 (m).

33 Examples of modular forms Suppose q =exp(πiτ) so Im(τ) > 0 q < Let P(τ) = 4 n= nq n q n Q(τ) = + 40 P aτ+b cτ+d =(cτ + d) P(τ) πic(cτ + d). P(τ) is not a modular form n= n 3 q n q n mp(mτ) P(τ) is a modular form of weight and level m Q(τ) is a modular form of weight 4 (and level )

34 Ramanuan s alternative theories of elliptic functions z 4 = Q(τ) z = F ( 6, 5 6 ;;x ) z = P(τ) P(τ) z = F ( 4, 3 4 ;;x ) z 3 = z 4 = 3P(3τ) P(τ) 4P(4τ) P(τ) 3 z 3 = F ( 3, 3 ;;x 3) z 4 = F (, ;;x 4) q dx m dq = z mx m ( x m ), x m (e π/ m )=

35 Acommonwayofviewingall4theories f (4) := 4P(q4 ) P(q) 3 f (3) := 3P(q3 ) P(q) = = f () := P(q ) P(q) = f () := Q(q) / = P(q) = 4 = 3 η 4 η 4 4 η 4 f (4) η η3 3 f (3) η η 4 f () 6 η f () q q, Q(q) = q q η m = q m/4 ( q m ) = =

36 Outline Sporadic sequences Background: classical theory 3 Ramanuan s alternative theories 4 Beyond Ramanuan s alternative theories

37 F. Beukers, 983 (k + ) s k+ = (k + k + 3)s k + k s k, s 0 = z = s k x k k=0 x = q z = = = ( q 5 4 ) 5 ( q 5 ) 5 ( q 5 3 ) 5 ( q 5 ) 5 = r 5(q) 5 ( q ) ( q 5 4 ) 5 ( q 5 ) 5 How does it fit in with Jacobi and Ramanuan s theories?

38 Analogues of Clausen s formula Chan, Tanigawa, Yang and Zudilin (0): Clausen ( + cw ) u w = w( aw cw ) u ( + cw ) Almkvist, van Straten and Zudilin (0): Clausen ( aw cw ) u w = w t aw cw ( + ) u + =(a + a + b)u + c u ( + ) 3 t + = ( + )(a + a + a b)t (4c + a ) 3 t u 0 = t 0 =

39 Higher levels: Clausen example f (4) := 4P(q4 ) P(q) 3 = 3 η 4 η 4 4 η 4 f (4) f (5) := 5P(q5 ) P(q) 4 = k s = k=0 k=0 + k k + k k k η η5 f (5) ( + ) s + = ( + + 3)s + s R. Apéry: ζ() Q

40 Rogers-Ramanuan continued fraction 5P(q 5 ) P(q) 4 = k=0 k + k k η η5 f (5) r = r(q) = + + q /5 q q + q3 +. η η5 = r 5 ( r 5 r 0 ) f (5) ( + r 0 ).

41 Higher levels: Clausen example f (4) := 4P(q4 ) P(q) 3 = 3 η 4 η 4 4 η 4 f (4) f (6) := 30P(q6 ) 3P(q 3 )+P(q ) 5P(q) 4 + k η η η 3 η 6 = k k f (6) k=0 ( +) 3 t + =( +)( )t 3 t : Apéry, ζ(3) Q P(q) = 4 = q q, η m = q m/4 ( q m ) =

42 Summary, so far Levels,, 3: Ramanuan s theories to alternative bases Level 4: Jacobi Levels 5, 6, 6, 6, 8, 9: Zagier s sporadic sequences F functions correspond to weight one modular forms. (e.g., elliptic integral sum of two squares) ( + ) u + =(a + a + b)u + c u 3F functions correspond to weight two modular forms. ( + ) 3 t + = ( + )(a + a + a b)t (4c + a ) 3 t ( + ) 3 s + = ( + )(a + a + b)s +4c(4 )s

43 Other three-term recurrence relations s = k=0 4, level 0 k ( + ) 3 s + =( + )( )s + (64 4)s, Franel Experimental search: ( + ) 3 s + =( + )(a + a + b)s + (c + d)s (a, b, c, d) (3, 4, 7, 3) (6,, 64, 4) (4, 6, 9, ) level 7 0 8

44 Level 7 f (7) = 7P(q7 ) P(q) = q +k+k 6 = k= / k k η = η 3/ 7. k f (7) Level 0 k=0 f (0) := 0P(q0 )+5P(q 5 ) P(q ) P(q) 4 η η η 5 η 4/3 0 = k f (0) k=0

45 Level 3. Joint work with Dongxi Ye (x, x,...,x m ; q) = ( x q )( x q ) ( x m q ) R = R(q) =q ( q ) ( 3) (q, q 3, q 4, q 9, q 0, q ; q 3 ) = q (q, q 5, q 6, q 7, q 8, q ; q 3 ) = = r 5 (q) =q ( q ) 5( 5) (q, q 4 ; q 5 ) 5 = q (q, q 3 ; q 5 ) 5 = R 3 R = q = q = = ( q ) ( q 3 ) ( q ) 6 ( q 5 ) 6

46 Factorizations: Rogers-Ramanuan continued fraction = q = ( q ) 6 ( q 5 ) 6 = α 5 β 5 α 5 = β 5 = (q 5 s) / (q 5 s) / = = ( ζq ) 5 ( ζ 4 q ) 5 ( ζ q ) 5 ( ζ 3 q ) 5. α = 5, β = + 5, ζ =exp(πi/5), s = η6 5 η 6

47 Factorizations: level 3 R 3 R = ( q ) q ( q 3 ) = γ R R R 3 R = R δ R γ R = R (qs) /4 (ξq,ξ 3 q,ξ 4 q,ξ 9 q,ξ 0 q,ξ q; q) δ R = R (qs) /4 (ξ q,ξ 5 q,ξ 6 q,ξ 7 q,ξ 8 q,ξ. q; q) γ = 3 3, δ = 3+ 3, ξ =exp(πi/3), S = η 3 η

48 (q, q 5, q 6, q 7, q 8, q, q 3, q 3 ; q 3 ) γ q (q, q 3, q 4, q 9, q 0, q, q 3, q 3 ; q 3 ) =(ξ q,ξ 5 q,ξ 6 q,ξ 7 q,ξ 8 q,ξ q, q, q; q) (q, q 5, q 6, q 7, q 8, q, q 3, q 3 ; q 3 ) δ q (q, q 3, q 4, q 9, q 0, q, q 3, q 3 ; q 3 ) =(ξq,ξ 3 q,ξ 4 q,ξ 9 q,ξ 0 q,ξ q, q, q; q). γ = 3 3, δ = 3+ 3, ξ =exp(πi/3)

49 Weight modular functions q d dq log = a(n) = n n n ( ) a(n) ( + ) n=0 n n + The coefficients a(n) satisfy a 3-term recurrence relation.. n q d dq log R 3R R = R( 3R R ) A(n) ( + R ) n=0 The coefficients A(n) satisfy a 6-term recurrence relation. n

50 Degree 3 hypergeometric transformation formulas F, 5 ; ; 78x (+5x+3x )(+47x+3380x +5379x x 4 ) x x x x 4 = F, 5 ; ; 78x 3 (+5x+3x )(+7x+0x +9x 3 +x 4 ) x + 0x + 9x 3 + x 4 (+5x+3x )(+47x+3380x +5379x x 4 ) 3 3F 6, 5 6, 78x ;, ; + 47x x x x 4 = 3 F 6, 5 6, ;, ; 78x 3 (+5x+3x )(+7x+0x +9x 3 +x 4 ) 3 +7x + 0x + 9x 3 + x 4.

51 Other levels similar to 5 and 3: ( ) 4 If ( ) 4, let f () = P(q ) P(q). f () = A ()x where A () satisfies a recurrence relation, of order given by: order

52 Level. Joint work with J. Ge and D. Ye f (3) = 3P(q3 ) P(q) f (7) = 7P(q7 ) P(q) 6 f () = = = = k= = k= = k= q +k+k q +k+k q +k+3k f () = η A () η /(+), =3, 7, f () ( + ) 3 A ( + ) = ( + )( )A () 8(7 + )A ( ) + ( )( )A ( ).

53 Levels 4 and 5. Joint work with Dongxi Ye Similar results, because: d = = 4 d 4 d = = 4 d 5 Cubic transformation of a level 5 function f (x) = n=0 n n n n n + x n = c n x n n=0 +9x + 7x f x ( + 9x + 7x ) = +3x +3x f x 3 ( + 3x +3x )

54 Quintic transformation of a level 3 function g(x) = a n x n, a 0 = n=0 (n + ) 3 a n+ =(n + )(7n +7n + 3)a n n(9n + 4)a n + 30n(n )(n )a n x = v ( + 3v) = w +w w near x =0 vw xg(x) = 3v( + w ) w( 9v ) 3 F = 5vw 3v( + w )+w( 9v ) 3 F 3,, 3, 3,, 3, ; ; 08v 3 w (w + 7v 3 ) 08v 3 w ( + 7v 3 w ).

55 Jacobi, level 4 q 4 = ( ) q = = = Level 5 analogue q +k+k = k= = k= =3 q +k+4k +3 = k= = k= q (+ ) q 5 +5k+5k 4 q +k+k

56 Other levels Level 0 t(k) = k k 4, Franel, 895, four-term recurrence κ = r(q)r (q ( q 0 9 )( q 0 8 )( q 0 )( q 0 ) )=q ( q 0 7 )( q 0 6 )( q 0 4 )( q 0 3 ) Level = κ = q = ( q )( q ) ( q 5 )( q 7 ) The coefficients satisfy a six-term recurrence relation. There are also results for levels 8, 0, 4, 3

57 Summary Table: Order of recurrence relations for level , , 4, 3 5 order Levels,, 3: Ramanuan s alternative theories Level 4: Jacobi Levels 5, 6, 8, 9: Zagier s sporadic sequences Level 5: Rogers-Ramanuan cont. frac. Weight : Apéry ζ() Level 6: 3 theories, one involves the cubic continued fraction Weight : Apéry ζ(3), Domb, Almkvist-Zudilin numbers Levels, 3, 4, 5, 7, 9, 3, 5: ( ) 4 share a common theory Levels 3, 7, : another common theory Levels 4, 5: very similar theories

58 Ramanuan s series for /π Ramanuan-Gosper, level, degree 9 π = k 4k ( k) 980 k k 396 4k. k=0 η 4 η P(q = ) P(q) q=exp( π 9/) Another example, level, degree 7 π = ( ) k (67 + k) c(k) k k=0 where c(k) satisfies a 4-term recurrence relation. η η k q +k+3k = 44 q= exp( π 7/)

59 Another example, with an 80 in it Zagier s sporadic sequence: (a, b, c) = (0, 3, 9) Level: =6 Degree: N =5 π = 3 7/ k=0 k k k k (3 + 80k) 8 k x := 4η η η 3 η 6 6P 6 3P 3 +P P q=exp( π N/) = 8 N 4ax 6cx = /. The end!