Supercongruences for Apéry-like numbers

Transcription

1 Supercongruences for Apéry-lie numbers npr 2 seminar NIE, Singapore August 13, 2014 University of Illinois at Urbana Champaign A(n) = n ( n ) 2 ( n + =0 1, 5, 73, 1445, 33001, , ,... ) 2 Jon Borwein Dir Nuyens James Wan Wadim Zudilin Robert Osburn Brundaban Sahu Mathew Rogers U. of Newcastle, AU K.U.Leuven, BE SUTD, SG U. of Newcastle, AU U. of Dublin, IE NISER, IN U. of Montreal, CA 1 / 45

2 Apéry numbers and the irrationality of ζ(3) The Apéry numbers 1, 5, 73, 1445,... n ( ) n 2 ( ) n + 2 A(n) = satisfy =0 (n + 1) 3 u n+1 = (2n + 1)(17n n + 5)u n n 3 u n 1. 2 / 45

3 Apéry numbers and the irrationality of ζ(3) The Apéry numbers 1, 5, 73, 1445,... n ( ) n 2 ( ) n + 2 A(n) = satisfy =0 (n + 1) 3 u n+1 = (2n + 1)(17n n + 5)u n n 3 u n 1. THM Apéry 78 ζ(3) = n=1 1 n 3 is irrational. proof The same recurrence is satisfied by the near -integers n ( ) n 2 ( ) n + 2 n B(n) = 1 j 3 + ( 1) m 1 2m 3( n m =0 j=1 m=1 )( n+m m ). Then, B(n) A(n) ζ(3). But too fast for ζ(3) to be rational. 2 / 45

4 Zagier s search and Apéry-lie numbers Recurrence for Apéry numbers is the case (a, b, c) = (17, 5, 1) of Q Beuers, Zagier (n + 1) 3 u n+1 = (2n + 1)(an 2 + an + b)u n cn 3 u n 1. Are there other tuples (a, b, c) for which the solution defined by u 1 = 0, u 0 = 1 is integral? 3 / 45

5 Zagier s search and Apéry-lie numbers Recurrence for Apéry numbers is the case (a, b, c) = (17, 5, 1) of Q Beuers, Zagier (n + 1) 3 u n+1 = (2n + 1)(an 2 + an + b)u n cn 3 u n 1. Are there other tuples (a, b, c) for which the solution defined by u 1 = 0, u 0 = 1 is integral? Essentially, only 14 tuples (a, b, c) found. 4 hypergeometric and 4 Legendrian solutions 6 sporadic solutions (Almvist Zudilin) Similar (and intertwined) story for: (n + 1) 2 u n+1 = (an 2 + an + b)u n cn 2 u n 1 (n + 1) 3 u n+1 = (2n + 1)(an 2 + an + b)u n n(cn 2 + d)u n 1 (Beuers, Zagier) (Cooper) 3 / 45

6 Apéry-lie numbers Hypergeometric and Legendrian solutions have generating functions 3F 2 ( 1 2, α, 1 α 1, 1 ) 4C αz, 1 1 C α z 2 F 1 with α = 1 2, 1 3, 1 4, 1 6 and C α = 2 4, 3 3, 2 6, The six sporadic solutions are: (a, b, c) (7, 3, 81) (11, 5, 125) (10, 4, 64) (12, 4, 16) (9, 3, 27) (17, 5, 1) A(n) ( 1) 3 n 3( n )( n+ ) (3)! 3 n! 3 ( 1)( ) ( n 3 (4n 5 1 ( n 3n ) 2 ( 2 )( 2(n ) ) ( n ) 2 ( 2 ) 2 n ( n 2 ( n,l ) l)( l ( n 2 ( n+ ) n n )( +l n ) 2 ) ) + ( 4n 5 3n ( ) α, 1 α 2 C α z 1, 1 C α z ) ) 4 / 45

7 Modularity of Apéry-lie numbers The Apéry numbers 1, 5, 73, 1145,... n ( ) n 2 ( ) n + 2 A(n) = satisfy =0 η 7 (2τ)η 7 (3τ) η 5 (τ)η 5 = ( η 12 (τ)η 12 ) n (6τ) A(n) (6τ) η 12 (2τ)η 12. (3τ) n 0 modular form O( 4 ) modular function O( 4 ) 5 / 45

8 Modularity of Apéry-lie numbers The Apéry numbers 1, 5, 73, 1145,... n ( ) n 2 ( ) n + 2 A(n) = satisfy =0 η 7 (2τ)η 7 (3τ) η 5 (τ)η 5 = ( η 12 (τ)η 12 ) n (6τ) A(n) (6τ) η 12 (2τ)η 12. (3τ) n 0 FACT modular form O( 4 ) modular function O( 4 ) Not at all evidently, such a modular parametrization exists for all nown Apéry-lie numbers! Context: f(τ) modular form of weight x(τ) modular function y(x) such that y(x(τ)) = f(τ) Then y(x) satisfies a linear differential euation of order / 45

9 Supercongruences for Apéry numbers Chowla, Cowles and Cowles (1980) conjectured that, for p 5, A(p) 5 (mod p 3 ). 6 / 45

10 Supercongruences for Apéry numbers Chowla, Cowles and Cowles (1980) conjectured that, for p 5, A(p) 5 (mod p 3 ). Gessel (1982) proved that A(mp) A(m) (mod p 3 ). 6 / 45

11 Supercongruences for Apéry numbers Chowla, Cowles and Cowles (1980) conjectured that, for p 5, A(p) 5 (mod p 3 ). Gessel (1982) proved that A(mp) A(m) (mod p 3 ). THM Beuers, Coster 85, 88 The Apéry numbers satisfy the supercongruence (p 5) A(mp r ) A(mp r 1 ) (mod p 3r ). 6 / 45

12 Supercongruences for Apéry numbers Chowla, Cowles and Cowles (1980) conjectured that, for p 5, A(p) 5 (mod p 3 ). Gessel (1982) proved that A(mp) A(m) (mod p 3 ). THM Beuers, Coster 85, 88 The Apéry numbers satisfy the supercongruence (p 5) A(mp r ) A(mp r 1 ) (mod p 3r ). EG Simple combinatorics proves the congruence ( ) 2p = ( )( ) p p (mod p 2 ). p p For p 5, Wolstenholme s congruence shows that, in fact, ( ) 2p 2 (mod p 3 ). p 6 / 45

13 Conjecturally, supercongruences lie A(mp r ) A(mp r 1 ) (mod p 3r ) Robert Osburn Brundaban Sahu U. of Dublin, IE NISER, IN hold for all Apéry-lie numbers. Osburn Sahu 09 Current state of affairs for the six sporadic seuences from earlier: (a, b, c) (7, 3, 81) (11, 5, 125) (10, 4, 64) (12, 4, 16) (9, 3, 27) (17, 5, 1) A(n) ( 1) 3 n 3( n )( n+ ) (3)! 3 n! 3 open!! modulo p 2 Amdeberhan 14 ( 1)( ) ( n 3 (4n 5 1 ) ( + 4n 5 ) ) Osburn Sahu S 14 ( n ( n ) 2 ( 2 n 3n 3n ) Chan Cooper Sica 10 n Osburn Sahu 11 ) 2 Osburn Sahu S 14 ) 2 ( 2 )( 2(n ) ( n 2 ( n )( +l ),l ) l)( l n open ( n 2 ( n+ ) 2 ) n Beuers, Coster / 45

14 Non-super congruences are abundant a(mp r ) a(mp r 1 ) (mod p r ) (C) realizable seuences a(n), i.e., for some map T : X X, a(n) = #{x X : T n x = x} points of period n Everest van der Poorten Puri Ward 02, Arias de Reyna 05 8 / 45

15 Non-super congruences are abundant a(mp r ) a(mp r 1 ) (mod p r ) (C) realizable seuences a(n), i.e., for some map T : X X, a(n) = #{x X : T n x = x} points of period n Everest van der Poorten Puri Ward 02, Arias de Reyna 05 a(n) = ct Λ(x) n van Straten Samol 09 if origin is only interior pt of the Newton polyhedron of Λ(x) Z p[x ±1 1,..., x±1 d ] 8 / 45

16 Non-super congruences are abundant a(mp r ) a(mp r 1 ) (mod p r ) (C) realizable seuences a(n), i.e., for some map T : X X, a(n) = #{x X : T n x = x} points of period n Everest van der Poorten Puri Ward 02, Arias de Reyna 05 a(n) = ct Λ(x) n van Straten Samol 09 if origin is only interior pt of the Newton polyhedron of Λ(x) Z p[x ±1 1,..., x±1 d ] ( ) a(n) n T n Z[[T ]]. n=1 This is a natural condition in formal group theory. If a(1) = 1, then (C) is euivalent to exp 8 / 45

17 Cooper s sporadic seuences Cooper s search for integral solutions to (n + 1) 3 u n+1 = (2n + 1)(an 2 + an + b)u n n(cn 2 + d)u n 1 revealed three additional sporadic solutions: s 7 (n) = n =0 =0 ( n ) 2 ( n + )( ) 2 n s 10 (n) = n s 10 and supercongruence nown ) 4 [n/3] ( )( )( ) [( ) ( )] n 2 2(n ) 2n 3 1 2n 3 s 18 (n) = ( 1) + n n n =0 ( n 9 / 45

18 Cooper s sporadic seuences Cooper s search for integral solutions to (n + 1) 3 u n+1 = (2n + 1)(an 2 + an + b)u n n(cn 2 + d)u n 1 revealed three additional sporadic solutions: s 7 (n) = n =0 =0 ( n ) 2 ( n + )( ) 2 n s 10 (n) = n s 10 and supercongruence nown ) 4 [n/3] ( )( )( ) [( ) ( )] n 2 2(n ) 2n 3 1 2n 3 s 18 (n) = ( 1) + n n n =0 ( n CONJ Cooper 2012 s 7 (mp) s 7 (m) (mod p 3 ) p 3 s 18 (mp) s 18 (m) (mod p 2 ) 9 / 45

19 Cooper s sporadic seuences Cooper s search for integral solutions to (n + 1) 3 u n+1 = (2n + 1)(an 2 + an + b)u n n(cn 2 + d)u n 1 revealed three additional sporadic solutions: s 7 (n) = n =0 =0 ( n ) 2 ( n + )( ) 2 n s 10 (n) = n s 10 and supercongruence nown ) 4 [n/3] ( )( )( ) [( ) ( )] n 2 2(n ) 2n 3 1 2n 3 s 18 (n) = ( 1) + n n n =0 ( n CONJ Cooper 2012 THM Osburn- Sahu-S 2014 s 7 (mp) s 7 (m) (mod p 3 ) p 3 s 18 (mp) s 18 (m) (mod p 2 ) s 7 (mp r ) s 7 (mp r 1 ) (mod p 3r ) p 5 s 18 (mp r ) s 18 (mp r 1 ) (mod p 2r ) 9 / 45

20 Applications of Apéry-lie numbers Random wals Jon Borwein Dir Nuyens James Wan Wadim Zudilin U. of Newcastle, AU K.U.Leuven, BE SUTD, SG U. of Newcastle, AU Series for 1/π Mat Rogers U. of Montreal, CA Positivity of rational functions Wadim Zudilin U. of Newcastle, AU 10 / 45

21 Example I: Random wals n steps in the plane (length 1, random direction) 11 / 45

22 Example I: Random wals n steps in the plane (length 1, random direction) 11 / 45

23 Example I: Random wals n steps in the plane (length 1, random direction) 11 / 45

24 Example I: Random wals n steps in the plane (length 1, random direction) 11 / 45

25 Example I: Random wals n steps in the plane (length 1, random direction) 11 / 45

26 Example I: Random wals n steps in the plane (length 1, random direction) 11 / 45

27 Example I: Random wals n steps in the plane (length 1, random direction) d 11 / 45

28 Example I: Random wals n steps in the plane (length 1, random direction) p n (x) probability density of distance traveled p 3 (x) p 4 (x) d p 5 (x) p 6 (x) / 45

29 Example I: Random wals n steps in the plane (length 1, random direction) p n (x) probability density of distance traveled p 3 (x) p 4 (x) d p 5 (x) p 6 (x) W n (s) = 0 x s p n (x) dx probability moments W 2 (1) = 4 π, W 3(1) = 3 2 1/3 16 π 4 Γ6 ( 1 3 ) /3 ( ) 2 4 π 4 Γ6 3 classical Borwein Nuyens S Wan, / 45

30 Example I: Random wals The probability moments W n (s) = include the Apéry-lie numbers W 3 (2) = W 4 (2) = j=0 j=0 ( j ( j 0 ) 2 ( 2j j ) 2 ( 2j j x s p n (x) dx ), )( ) 2( j). j 12 / 45

31 Example I: Random wals The probability moments W n (s) = include the Apéry-lie numbers W 3 (2) = W 4 (2) = j=0 j=0 ( j ( j 0 ) 2 ( 2j j ) 2 ( 2j j x s p n (x) dx ), )( ) 2( j). j THM Borwein- Nuyens- S-Wan 2010 W n (2) = a 1 + +a n= ( a 1,..., a n ) 2 12 / 45

32 Example I: Random wals p 2 (x) p 3 (x) p 4 (x) 0.05 p 5 (x) 2 p 2 (x) = π 4 x 2 p 3 (x) = 2 3 π x (3 + x 2 ) 2 F 1 p 4 (x) = 2 16 x 2 π 2 x p 5(0) = ( 1 3, Re 3 F 2 ( 1 2, 1 2, , 7 6 x 2 ( 9 x 2) ) 2 (3 + x 2 ) 3 ( 16 x 2 ) ) 3 108x π 4 Γ( 1 15 )Γ( 2 15 )Γ( 4 15 )Γ( 8 15 ) easy classical with a spin new BSWZ / 45

33 Example II: Series for 1/π 4 π = n=0 16 π = n=0 (1/2) 3 n n! 3 (6n + 1) 1 4 n (1/2) 3 n n! 3 (42n + 5) 1 2 6n Srinivasa Ramanujan Modular euations and approximations to π Quart. J. Math., Vol. 45, p , / 45

34 Example II: Series for 1/π 4 π = n=0 16 π = n=0 1 π = (1/2) 3 n n! 3 (6n + 1) 1 4 n (1/2) 3 n n! 3 (42n + 5) 1 n=0 2 6n (4n)! n n! n Last series used by Gosper in 1985 to compute 17, 526, 100 digits of π First proof of all of Ramanujan s 17 series by Borwein brothers Srinivasa Ramanujan Modular euations and approximations to π Quart. J. Math., Vol. 45, p , 1914 Jonathan M. Borwein and Peter B. Borwein Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity Wiley, / 45

35 Example II: Series for 1/π Sato observed that series for 1 π can be built from Apéry-lie numbers: EG Chan- Chan-Liu 2003 For the Domb numbers D(n) = 8 3π = n=0 n =0 ( n ) 2 ( 2 D(n) 5n n. )( 2(n ) n ), 15 / 45

36 Example II: Series for 1/π Sato observed that series for 1 π can be built from Apéry-lie numbers: EG Chan- Chan-Liu 2003 For the Domb numbers D(n) = 8 3π = n=0 n =0 ( n ) 2 ( 2 D(n) 5n n. )( 2(n ) n ), Sun offered a $520 bounty for a proof the following series: THM Rogers-S π = n=0 1054n n ( 2n n ) n =0 ( n ) 2 ( 2 n ) ( 1) 8 2 n 15 / 45

37 Example II: Series for 1/π Suppose we have a seuence a n with modular parametrization a n x(τ) n = f(τ). Then: n=0 modular function n=0 (1/2) 3 n n! 3 (42n + 5) 1 16 = 26n π n=0 modular form a n (A + Bn)x(τ) n = Af(τ) + B x(τ) x (τ) f (τ) 16 / 45

38 Example II: Series for 1/π Suppose we have a seuence a n with modular parametrization a n x(τ) n = f(τ). Then: FACT n=0 modular function n=0 (1/2) 3 n n! 3 (42n + 5) 1 16 = 26n π n=0 modular form a n (A + Bn)x(τ) n = Af(τ) + B x(τ) x (τ) f (τ) For τ Q( d), x(τ) is an algebraic number. f (τ) is a uasimodular form. Prototypical E 2 (τ) satisfies τ 2 ( ) E 2 1 τ E2 (τ) = 6 πiτ. These are the main ingredients for series for 1/π. Mix and stir. 16 / 45

39 Example III: Positivity of rational functions A rational function F (x 1,..., x d ) = n 1,...,n d 0 is positive if a n1,...,n d > 0 for all indices. a n1,...,n d x n 1 1 xn d d EG The following rational functions are positive. Szegő 33 1 Kaluza 33 S(x, y, z) = 1 (x + y + z) (xy + yz + zx) A(x, y, z) = 1 1 (x + y + z) + 4xyz Asey Gasper 72 S 08 Asey Gasper 77 Koornwinder 78 Ismail Tamhanar 79 Gillis Reznic Zeilberger 83 Both functions are on the boundary of positivity. 17 / 45

40 Example III: Positivity of rational functions A rational function F (x 1,..., x d ) = n 1,...,n d 0 is positive if a n1,...,n d > 0 for all indices. a n1,...,n d x n 1 1 xn d d EG The following rational functions are positive. Szegő 33 1 Kaluza 33 S(x, y, z) = 1 (x + y + z) (xy + yz + zx) A(x, y, z) = 1 1 (x + y + z) + 4xyz Asey Gasper 72 S 08 Asey Gasper 77 Koornwinder 78 Ismail Tamhanar 79 Gillis Reznic Zeilberger 83 Both functions are on the boundary of positivity. The diagonal coefficients of A are the Franel numbers n =0 ( ) n / 45

41 Example III: Positivity of rational functions CONJ Kauers- Zeilberger 2008 The following rational function is positive: 1 1 (x + y + z + w) + 2(yzw + xzw + xyw + xyz) + 4xyzw. Would imply conjectured positivity of Lewy Asey rational function 1 1 (x + y + z + w) (xy + xz + xw + yz + yw + zw). Recent proof of non-negativity by Scott and Soal, / 45

42 Example III: Positivity of rational functions CONJ Kauers- Zeilberger 2008 The following rational function is positive: 1 1 (x + y + z + w) + 2(yzw + xzw + xyw + xyz) + 4xyzw. Would imply conjectured positivity of Lewy Asey rational function 1 1 (x + y + z + w) (xy + xz + xw + yz + yw + zw). Recent proof of non-negativity by Scott and Soal, 2013 PROP S-Zudilin 2013 The Kauers Zeilberger function has diagonal coefficients d n = n ( ) n 2 ( ) 2 2. n =0 18 / 45

43 Example III: Positivity of rational functions Consider rational functions F = 1/p(x 1,..., x d ) with p a symmetric polynomial, linear in each variable. Q Under what condition(s) is the positivity of F implied by the positivity of its diagonal? EG 1 1+x+y has positive diagonal coefficients but is not positive. 19 / 45

44 Example III: Positivity of rational functions Consider rational functions F = 1/p(x 1,..., x d ) with p a symmetric polynomial, linear in each variable. Q Under what condition(s) is the positivity of F implied by the positivity of its diagonal? EG 1 1+x+y has positive diagonal coefficients but is not positive. Q F positive diagonal of F and F xd =0 are positive? 19 / 45

45 Example III: Positivity of rational functions Consider rational functions F = 1/p(x 1,..., x d ) with p a symmetric polynomial, linear in each variable. Q Under what condition(s) is the positivity of F implied by the positivity of its diagonal? EG 1 1+x+y has positive diagonal coefficients but is not positive. Q F positive diagonal of F and F xd =0 are positive? THM S-Zudilin 2013 F (x, y) = c 1 (x + y) + c 2 xy diagonal of F and F xd =0 are positive is positive 19 / 45

46 Multivariate Apéry numbers 20 / 45

47 Apéry numbers as diagonals Given a series F (x 1,..., x d ) = n 1,...,n d 0 a(n 1,..., n d )x n 1 1 xn d d, its diagonal coefficients are the coefficients a(n,..., n). THM Gessel, Zeilberger, Lipshitz The diagonal of a rational function is D-finite. More generally, the diagonal of a D-finite function is D-finite. F K[[x 1,..., x d ]] is D-finite if its partial derivatives span a finitedimensional vector space over K(x 1,..., x d ). 21 / 45

48 Apéry numbers as diagonals Given a series F (x 1,..., x d ) = n 1,...,n d 0 a(n 1,..., n d )x n 1 1 xn d d, its diagonal coefficients are the coefficients a(n,..., n). THM Gessel, Zeilberger, Lipshitz The diagonal of a rational function is D-finite. More generally, the diagonal of a D-finite function is D-finite. F K[[x 1,..., x d ]] is D-finite if its partial derivatives span a finitedimensional vector space over K(x 1,..., x d ). EG Christol 1984 The Apéry numbers are the diagonal coefficients of 1 (1 x 1 ) [(1 x 2 )(1 x 3 )(1 x 4 )(1 x 5 ) x 1 x 2 x 3 ]. Such identities are routine to prove, but much harder to discover. 21 / 45

49 Apéry numbers as diagonals THM S 2013 The Apéry numbers are the diagonal coefficients of 1 (1 x 1 x 2 )(1 x 3 x 4 ) x 1 x 2 x 3 x 4 22 / 45

50 Apéry numbers as diagonals THM S 2013 The Apéry numbers are the diagonal coefficients of 1 (1 x 1 x 2 )(1 x 3 x 4 ) x 1 x 2 x 3 x 4 THM MacMahon 1915 For x = (x 1,..., x n ) and m = (m 1,..., m n ) Z n 0, [x m 1 n n ] det (I n BX) = [xm ] B i,j x j i=1 j=1 mi, where B C n n and X is the diagonal matrix with entries x 1,..., x n. 22 / 45

51 Apéry numbers as diagonals THM S 2013 THM MacMahon 1915 The Apéry numbers are the diagonal coefficients of 1 = (1 x 1 x 2 )(1 x 3 x 4 ) x 1 x 2 x 3 x 4 n Z 4 0 For x = (x 1,..., x n ) and m = (m 1,..., m n ) Z n 0, [x m 1 n n ] det (I n BX) = [xm ] B i,j x j i=1 j=1 A(n)x n. mi, where B C n n and X is the diagonal matrix with entries x 1,..., x n. B = , X = x 1 x 2 x 3 x 4 A(n) = [x n ](x 1 + x 2 + x 3 ) n 1 (x 1 + x 2 ) n 2 (x 3 + x 4 ) n 3 (x 2 + x 3 + x 4 ) n 4 22 / 45

52 Apéry numbers as diagonals THM S 2013 The Apéry numbers are the diagonal coefficients of 1 = (1 x 1 x 2 )(1 x 3 x 4 ) x 1 x 2 x 3 x 4 n Z 4 0 A(n)x n. The coefficients are the multivariate Apéry numbers A(n) = Z ( n1 )( n3 )( )( n1 + n 2 n3 + n 4 n 1 n 3 ). 22 / 45

53 Apéry numbers as diagonals THM S 2013 The Apéry numbers are the diagonal coefficients of 1 = (1 x 1 x 2 )(1 x 3 x 4 ) x 1 x 2 x 3 x 4 n Z 4 0 A(n)x n. The coefficients are the multivariate Apéry numbers A(n) = Z ( n1 )( n3 Univariate generating function: A(n)x n = n 0 where z = 1 34x + x 2. )( )( n1 + n 2 n3 + n 4 n 1 17 x z 4 2(1 + x + z) 3/2 3 F 2 ( 1 2, 1 2, 1 2 1, 1 n 3 ). 1024x (1 x + z) 4 ), 22 / 45

54 Apéry numbers as diagonals THM S 2013 The Apéry numbers are the diagonal coefficients of 1 = (1 x 1 x 2 )(1 x 3 x 4 ) x 1 x 2 x 3 x 4 n Z 4 0 A(n)x n. Well-developed theory of multivariate asymptotics e.g., Pemantle Wilson Such diagonals are algebraic modulo p r. Furstenberg, Deligne 67, 84 Automatically leads to congruences such as { 1 (mod 8), if n even, A(n) 5 (mod 8), if n odd. Chowla Cowles Cowles 80 Rowland Yassawi / 45

55 Multivariate supercongruences THM S 2013 Define A(n) = A(n 1, n 2, n 3, n 4 ) by 1 = (1 x 1 x 2 )(1 x 3 x 4 ) x 1 x 2 x 3 x 4 n Z 4 0 The Apéry numbers are the diagonal coefficients. A(n)x n. For p 5, we have the multivariate supercongruences A(np r ) A(np r 1 ) (mod p 3r ). 23 / 45

56 Multivariate supercongruences THM S 2013 Define A(n) = A(n 1, n 2, n 3, n 4 ) by 1 = (1 x 1 x 2 )(1 x 3 x 4 ) x 1 x 2 x 3 x 4 n Z 4 0 The Apéry numbers are the diagonal coefficients. A(n)x n. For p 5, we have the multivariate supercongruences A(np r ) A(np r 1 ) (mod p 3r ). a(n)x n = F (x) = a(pn)x pn = 1 p n 0 n 0 p 1 F (ζp x) =0 ζ p = e 2πi/p Hence, both A(np r ) and A(np r 1 ) have rational generating function. The proof, however, relies on an explicit binomial sum for the coefficients. 23 / 45

57 Multivariate supercongruences THM S 2013 Define A(n) = A(n 1, n 2, n 3, n 4 ) by 1 = (1 x 1 x 2 )(1 x 3 x 4 ) x 1 x 2 x 3 x 4 n Z 4 0 The Apéry numbers are the diagonal coefficients. A(n)x n. For p 5, we have the multivariate supercongruences A(np r ) A(np r 1 ) (mod p 3r ). By MacMahon s Master Theorem, A(n) = ( )( )( )( ) n1 n3 n1 + n 2 n3 + n 4. n 1 n 3 Z 23 / 45

58 Multivariate supercongruences THM S 2013 Define A(n) = A(n 1, n 2, n 3, n 4 ) by 1 = (1 x 1 x 2 )(1 x 3 x 4 ) x 1 x 2 x 3 x 4 n Z 4 0 The Apéry numbers are the diagonal coefficients. A(n)x n. For p 5, we have the multivariate supercongruences A(np r ) A(np r 1 ) (mod p 3r ). By MacMahon s Master Theorem, A(n) = ( )( )( )( ) n1 n3 n1 + n 2 n3 + n 4. n 1 n 3 Z Because A(n 1) = A( n, n, n, n), we also find A(mp r 1) A(mp r 1 1) (mod p 3r ). Beuers / 45

59 Many more conjectural multivariate supercongruences Exhaustive search by Alin Bostan and Bruno Salvy: 1/(1 p(x, y, z, w)) with p(x, y, z, w) a sum of distinct monomials; Apéry numbers as diagonal 1 1 (x + y + xy)(z + w + zw) 1 1 (1 + w)(z + xy + yz + zx + xyz) 1 1 (y + z + xy + xz + zw + xyw + xyzw) 1 1 (y + z + xz + wz + xyw + xzw + xyzw) 1 1 (z + xy + yz + xw + xyw + yzw + xyzw) 1 1 (z + (x + y)(z + w) + xyz + xyzw) 24 / 45

60 Many more conjectural multivariate supercongruences Exhaustive search by Alin Bostan and Bruno Salvy: 1/(1 p(x, y, z, w)) with p(x, y, z, w) a sum of distinct monomials; Apéry numbers as diagonal 1 1 (x + y + xy)(z + w + zw) 1 1 (1 + w)(z + xy + yz + zx + xyz) 1 1 (y + z + xy + xz + zw + xyw + xyzw) 1 1 (y + z + xz + wz + xyw + xzw + xyzw) 1 1 (z + xy + yz + xw + xyw + yzw + xyzw) 1 1 (z + (x + y)(z + w) + xyz + xyzw) CONJ S 2014 The coefficients B(n) of each of these satisfy, for p 5, B(np r ) B(np r 1 ) (mod p 3r ). 24 / 45

61 A simple conjectural tip of an iceberg CONJ S 2013 The coefficients F (n) of 1 = 1 (x 1 + x 2 + x 3 ) + 4x 1 x 2 x 3 n Z 3 0 F (n)x n satisfy, for p 5, the multivariate supercongruences F (np r ) F (np r 1 ) (mod p 3r ). Here, the diagonal coefficients are the Franel numbers n ( ) n 3 F (n) =. =0 25 / 45

62 A simple conjectural tip of an iceberg CONJ S 2013 The coefficients F (n) of 1 = 1 (x 1 + x 2 + x 3 ) + 4x 1 x 2 x 3 n Z 3 0 F (n)x n satisfy, for p 5, the multivariate supercongruences F (np r ) F (np r 1 ) (mod p 3r ). Here, the diagonal coefficients are the Franel numbers n ( ) n 3 F (n) =. The Franel numbers also appear as the diagonal coefficients of =0 1 (1 x 1 )(1 x 2 )(1 x 3 ) x 1 x 2 x 3, for which we can prove the above multivariate supercongruences. 25 / 45

63 Extended version of main result THM S 2014 Let λ = (λ 1,..., λ l ) Z l >0 with d = λ λ l, and set s(j) = λ λ j 1. Define A λ (n) by λ l j 1 j=1 r=1 x s(j)+r x 1 x 2 x d 1 = n Z d 0 A λ (n)x n. If l 2, then, for all primes p and integers r 1, A λ (np r ) A λ (np r 1 ) (mod p 2r ). If l 2 and max(λ 1,..., λ l ) 2, then, for primes p 5 and integers r 1, A λ (np r ) A λ (np r 1 ) (mod p 3r ). 26 / 45

64 Further examples EG The Apéry-lie numbers, associated with ζ(2), B(n) = n =0 ( n ) 2 ( n + are the diagonal coefficients of the rational function ), 1 = (1 x 1 x 2 )(1 x 3 ) x 1 x 2 x 3 n Z 3 0 B(n)x n. COR We find and, for primes p 5, B(n) = Z ( n1 )( n1 + n 2 n 1 )( n3 B(p r n) B(p r 1 n) (mod p 3r ). ), The diagonal case recovers supercongruences of Coster, / 45

65 Further examples EG The numbers Y d (n) = n =0 ( ) n d, are the diagonal coefficients of the rational function 1 = (1 x 1 )(1 x 2 ) (1 x d ) x 1 x 2 x d d = 3: Franel, d = 4: Yang Zudilin n Z d 0 Y d (n)x n. COR We find Y d (n) = 0 ( n1 )( n2 ) ( ) nd, and, for d 2 and primes p 5, Y d (p r n) Y d (p r 1 n) (mod p 3r ). This generalizes a result of Chan Cooper Sica, / 45

66 -analogs 29 / 45

67 Basic -analogs The natural number n has the -analog: [n] = n 1 1 = n 1 In the limit 1 a -analog reduces to the classical object. 30 / 45

68 Basic -analogs The natural number n has the -analog: [n] = n 1 1 = n 1 In the limit 1 a -analog reduces to the classical object. The -factorial: [n]! = [n] [n 1] [1] The -binomial coefficient: ( ) n = ( ) [n]! n []! [n ]! = n D1 30 / 45

69 A -binomial coefficient EG ( ) 6 = = 3 5 ( ) 6 = ( )( ) / 45

70 A -binomial coefficient EG ( ) 6 = = 3 5 ( ) 6 = ( )( ) = (1 + 2 ) ( ) ( ) }{{}}{{} =[3] =[5] 31 / 45

71 A -binomial coefficient EG ( ) 6 = = 3 5 ( ) 6 = ( )( ) = (1 + 2 ) ( ) ( ) }{{}}{{}}{{} =Φ 6 () =[3] =[5] The cyclotomic polynomial Φ 6 () becomes 1 for = 1 and hence invisible in the classical world 31 / 45

72 The coefficients of -binomial coefficients Here s some -binomials in expanded form: EG ( ) 6 2 ( ) 9 3 = = The degree of the -binomial is (n ). All coefficients are positive! In fact, the coefficients are unimodal. Sylvester, / 45

73 -binomials: Pascal s triangle The -binomials can be build from the -Pascal rule: ( ) ( ) ( ) n n 1 n 1 = + 1 D2 33 / 45

74 -binomials: Pascal s triangle The -binomials can be build from the -Pascal rule: ( ) ( ) ( ) n n 1 n 1 = + 1 D (1 + ) (1 + ) EG ( ) 4 = ( ) = / 45

75 -binomials: combinatorial ( ) n = w(s) where w(s) = S ( n j ) w(s) = normalized sum of S s j j D3 EG {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4} 0 ( ) = / 45

76 -binomials: combinatorial ( ) n = w(s) where w(s) = S ( n j ) w(s) = normalized sum of S s j j D3 EG {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4} 0 ( ) = The coefficient of m in ( ) n counts the number of -element subsets of n whose normalized sum is m 34 / 45

77 -binomials: combinatorial ( ) n = w(s) where w(s) = S ( n j ) w(s) = normalized sum of S s j j D3 EG {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4} 0 ( ) = The coefficient of m in ( ) n counts the number of -element subsets of n whose normalized sum is m partitions λ of m whose Ferrer s diagram fits in a (n ) box 34 / 45

78 -Chu-Vandermonde Different representations mae different properties apparent! Chu-Vandermonde: ( ) m+n ( = m )( n ) j j j 35 / 45

79 -Chu-Vandermonde Different representations mae different properties apparent! Chu-Vandermonde: ( ) m+n ( = m )( n ) j j j Purely from the combinatorial representation: ( ) m + n = S (+1)/2 S ( m+n ) 35 / 45

80 -Chu-Vandermonde Different representations mae different properties apparent! Chu-Vandermonde: ( ) m+n ( = m )( n ) j j j Purely from the combinatorial representation: ( ) m + n = S (+1)/2 S ( m+n ) = j S 1 ( m j ) S 2 ( n j) S 1 + S 2 +( j)m (+1)/2 35 / 45

81 -Chu-Vandermonde Different representations mae different properties apparent! Chu-Vandermonde: ( ) m+n ( = m )( n ) j j j Purely from the combinatorial representation: ( ) m + n = S (+1)/2 S ( m+n ) = j = j S 1 ( m j ) S 2 ( j) n ( ) ( ) m n j j S 1 + S 2 +( j)m (+1)/2 (m j)( j) 35 / 45

82 -binomials: algebraic Let be a prime power. ( ) n = number of -dim. subspaces of F n D4 36 / 45

83 -binomials: algebraic Let be a prime power. ( ) n = number of -dim. subspaces of F n D4 Number of ways to choose linearly independent vectors in F n : ( n 1)( n ) ( n 1 ) 36 / 45

84 -binomials: algebraic Let be a prime power. ( ) n = number of -dim. subspaces of F n D4 Number of ways to choose linearly independent vectors in F n : ( n 1)( n ) ( n 1 ) Hence the number of -dim. subspaces of F n is: ( n 1)( n ) ( n 1 ) ( 1)( ) ( 1 ) = ( ) n 36 / 45

85 -binomials: noncommuting variables Suppose yx = xy where commutes with x, y. Then: (x + y) n = n j=0 ( ) n j x j y n j D5 37 / 45

86 -binomials: noncommuting variables Suppose yx = xy where commutes with x, y. Then: (x + y) n = n j=0 ( ) n x j y n j j D5 EG ( ) 4 x 2 y 2 = xxyy + xyxy + xyyx + yxxy + yxyx + yyxx 2 = ( )x 2 y 2 37 / 45

87 -binomials: noncommuting variables Suppose yx = xy where commutes with x, y. Then: (x + y) n = n j=0 ( ) n x j y n j j D5 EG ( ) 4 x 2 y 2 = xxyy + xyxy + xyyx + yxxy + yxyx + yyxx 2 = ( )x 2 y 2 Let X f(x) = xf(x) and Q f(x) = f(x). Then: QX f(x) = xf(x) = XQ f(x) 37 / 45

88 Summary: the -binomial coefficient The -binomial coefficient: ( ) n = Via a -version of Pascal s rule [n]! []! [n ]! Combinatorially, as the generating function of the element sums of -subsets of an n-set Algebraically, as the number of -dimensional subspaces of F n Via a binomial theorem for noncommuting variables Not touched here: analytical definition via -integral representations uantum groups arising in representation theory and physics 38 / 45

89 A -analog of Babbage s congruence Using -Chu-Vandermonde ( ) 2p = ( ) ( p p p p ) (p )2 p2 + 1 (mod [p] 2 ) Again, [p] divides ( ) p unless = 0 or = p. 39 / 45

90 A -analog of Babbage s congruence Using -Chu-Vandermonde ( ) 2p = ( ) ( p p p p ) (p )2 p2 + 1 (mod [p] 2 ) Again, [p] divides ( ) p unless = 0 or = p. ( ) THM 2p [2] p p 2 (mod [p]2 ) Actually, this argument extends to show: ( ) ( ) THM ap a Clar (mod [p] 2 bp b ) 1995 p2 39 / 45

91 A -analog of Babbage s congruence Using -Chu-Vandermonde ( ) 2p = ( ) ( p p p p ) (p )2 p2 + 1 (mod [p] 2 Similar results ) Again, [p] divides ( by ) Andrews; e.g.: p ( ) unless = 0 or = p. ( ) ap a ( ) (a b)b(p 2) (mod [p] 2 THM 2p bp [2] p p 2 (mod [p]2 ) b ) p George Andrews -analogs of the binomial coefficient congruences of Babbage, Wolstenholme and Glaisher Discrete Mathematics 204, 1999 Actually, this argument extends to show: ( ) ( ) THM ap a Clar (mod [p] 2 bp b ) 1995 p2 39 / 45

92 A -analog of Ljunggren s congruence The following answers the uestion of Andrews to find a -analog of Wolstenholme s congruence. THM S 2011 For primes p 5, ( ) ap bp ( ) ( a a b b + 1 p2 )( b ) p (p 1) 2 (mod [p] 3 ). 40 / 45

93 A -analog of Ljunggren s congruence The following answers the uestion of Andrews to find a -analog of Wolstenholme s congruence. THM S 2011 EG For primes p 5, ( ) ap bp ( ) ( a a b b + 1 p2 )( b ) p (p 1) 2 (mod [p] 3 ). Choosing p = 13, a = 2, and b = 1, we have ( ) 26 = ( 13 1) 2 + ( ) 3 f() 13 where f() = is an irreducible polynomial with integer coefficients. 40 / 45

94 Just coincidence? ( ) ap bp ( ) ( a a b b + 1 p2 )( b ) p (p 1) 2 (mod [p] 3 ) Ernst Jacobsthal (1952) proved that Ljunggren s classical congruence holds modulo p 3+r where r is the p-adic valuation of ( ) ( )( ) a a b + 1 ab(a b) = 2a. b b It would be interesting to see if this generalization has a nice analog in the -world. 41 / 45

95 The case of composite numbers ( ) ap bp ( ) ( a a b b + 1 p2 )( b ) p (p 1) 2 (mod [p] 3 ) Note that n is an integer if (n, 6) = / 45

96 The case of composite numbers ( ) ap bp ( ) ( a a b b + 1 p2 )( b ) p (p 1) 2 (mod [p] 3 ) Note that n is an integer if (n, 6) = 1. Ljunggren s -congruence holds modulo Φ n () 3 over integer coefficient polynomials if (n, 6) = 1 otherwise we get rational coefficients. EG n = 35, a = 2, b = 1 ( ) 70 = ( 35 1) 2 + Φ 35 () 3 f(), 35 where f() = Note that Φ 35 has degree / 45

97 The case of composite numbers ( ) ap bp ( ) ( a a b b + 1 p2 )( b ) p (p 1) 2 (mod [p] 3 ) Note that n is an integer if (n, 6) = 1. Ljunggren s -congruence holds modulo Φ n () 3 over integer coefficient polynomials if (n, 6) = 1 otherwise we get rational coefficients. EG n = 12, a = 2, b = 1 ( ) 24 = (12 1) ( ) 3 f(), }{{} Φ 12 () where f() = / 45

98 -supercongruences for the Apéry numbers THM S 2014 The -analog of the Apéry numbers, defined as A (n) = n ( n (n )2 =0 ) 2 ( n + ) 2, satisfy A (1) = , A(1) = 5 A (pn) A p 2 (n) p (p 1) 2 f(n) (mod [p] 3 ). 43 / 45

99 -supercongruences for the Apéry numbers THM S 2014 The -analog of the Apéry numbers, defined as A (n) = n ( n (n )2 =0 ) 2 ( n + ) 2, satisfy A (1) = , A(1) = 5 A (pn) A p 2 (n) p (p 1) 2 f(n) (mod [p] 3 ). The numbers f(n) can be expressed as 0, 5, 292, 13005, ,... f(n) = n ( ) n 2 ( ) n g(n, ), g(n, ) = (2n ) + (n + ) 2. =0 Similar congruences for other Apéry-lie numbers? 43 / 45

100 Some of many open problems Supercongruences for all Apéry-lie numbers proof for all of them uniform explanation multivariable extensions Apéry-lie numbers as diagonals find minimal rational functions extend supercongruences any structure? Many further uestions remain. is the nown list complete? higher-order analogs, Calabi Yau DEs reason for modularity modular supercongruences Beuers 87, Ahlgren Ono 00 -analogs... ( ) p 1 A a(p) (mod p 2 ), 2 a(n) n = η 4 (2τ)η 4 (4τ) n=1 44 / 45

101 THANK YOU! Slides for this tal will be available from my website: A. Straub Multivariate Apéry numbers and supercongruences of rational functions Preprint, 2014 R. Osburn, B. Sahu, A. Straub Supercongruences for sporadic seuences to appear in Proceedings of the Edinburgh Mathematical Society, 2014 A. Straub, W. Zudilin Positivity of rational functions and their diagonals to appear in Journal of Approximation Theory (special issue dedicated to Richard Asey), 2014 M. Rogers, A. Straub A solution of Sun s $520 challenge concerning 520/π International Journal of Number Theory, Vol. 9, Nr. 5, 2013, p J. Borwein, A. Straub, J. Wan, W. Zudilin (appendix by D. Zagier) Densities of short uniform random wals Canadian Journal of Mathematics, Vol. 64, Nr. 5, 2012, p A. Straub A -analog of Ljunggren s binomial congruence DMTCS Proceedings: FPSAC 2011, p / 45