Supercongruences for Apéry-like numbers

Transcription

1 Supercongruences for Apéry-like numbers AKLS seminar on Automorphic Forms Universität zu Köln March, 205 University of Illinois at Urbana-Champaign A(n) = n k=0 ( ) n 2 ( n + k k k ) 2, 5, 73, 445, 3300, 89005,... Includes joint work with: Robert Osburn (University of Dublin) Brundaban Sahu (NISER, India) / 27

2 Rough outline Introducing Apéry-like numbers they are integer solutions to certain three-term recurrences are all of them known? Apéry-like numbers have interesting properties connection to modular forms supercongruences (still open in several cases) multivariate extensions polynomial analogs Apéry-like numbers occur in interesting places (if time permits) moments of planar random walks series for /π positivity of rational functions counting points on algebraic varieties... 2 / 27

3 Apéry numbers and the irrationality of ζ(3) The Apéry numbers, 5, 73, 445,... n ( ) n 2 ( ) n + k 2 A(n) = k k satisfy k=0 (n + ) 3 A(n + ) = (2n + )(7n 2 + 7n + 5)A(n) n 3 A(n ). 3 / 27

4 Apéry numbers and the irrationality of ζ(3) The Apéry numbers, 5, 73, 445,... n ( ) n 2 ( ) n + k 2 A(n) = k k satisfy k=0 (n + ) 3 A(n + ) = (2n + )(7n 2 + 7n + 5)A(n) n 3 A(n ). THM Apéry 78 ζ(3) = n= n 3 is irrational. proof The same recurrence is satisfied by the near -integers n ( ) n 2 ( ) n + k 2 n B(n) = k k k j 3 + ( ) m 2m 3( n m k=0 j= m= )( n+m m ). Then, B(n) A(n) ζ(3). But too fast for ζ(3) to be rational. 3 / 27

5 Zagier s search and Apéry-like numbers Recurrence for Apéry numbers is the case (a, b, c) = (7, 5, ) of Q Beukers, Zagier (n + ) 3 u n+ = (2n + )(an 2 + an + b)u n cn 3 u n. Are there other tuples (a, b, c) for which the solution defined by u = 0, u 0 = is integral? 4 / 27

6 Zagier s search and Apéry-like numbers Recurrence for Apéry numbers is the case (a, b, c) = (7, 5, ) of Q Beukers, Zagier (n + ) 3 u n+ = (2n + )(an 2 + an + b)u n cn 3 u n. Are there other tuples (a, b, c) for which the solution defined by u = 0, u 0 = is integral? Essentially, only 4 tuples (a, b, c) found. (Almkvist Zudilin) 4 hypergeometric and 4 Legendrian solutions (with generating functions 3F 2 ( 2, α, α, ) 4C αz, with α = 2, 3, 4, 6 and C α = 2 4, 3 3, 2 6, ) 6 sporadic solutions ( ) α, α 2 C α z 2 F C α z, C α z Similar (and intertwined) story for: (n + ) 2 u n+ = (an 2 + an + b)u n cn 2 u n (n + ) 3 u n+ = (2n + )(an 2 + an + b)u n n(cn 2 + d)u n (Beukers, Zagier) (Cooper) 4 / 27

7 The six sporadic Apéry-like numbers (a, b, c) A(n) ( ) n 2 ( ) n + k 2 (7, 5, ) Apéry numbers k n k ( ) n 2 ( ) 2k 2 (2, 4, 6) k n k ( ) n 2 ( )( ) 2k 2(n k) (0, 4, 64) Domb numbers k k n k k ( )( ) n n + k (3k)! (7, 3, 8) ( ) k 3 n 3k Almkvist Zudilin numbers 3k n k! 3 k ( ) n 3 (( ) ( )) 4n 5k 4n 5k (, 5, 25) ( ) k + k 3n 3n k ( ) n 2 ( )( )( ) n k k + l (9, 3, 27) k l l n k,l 5 / 27

8 Apéry-like numbers and modular forms The Apéry numbers A(n) satisfy, 5, 73, 45,... η 7 (2τ)η 7 (3τ) η 5 (τ)η 5 = ( η 2 (τ)η 2 ) n (6τ) A(n) (6τ) η 2 (2τ)η 2. (3τ) n 0 modular form O( 4 ) modular function O( 4 ) = e 2πiτ 6 / 27

9 Apéry-like numbers and modular forms The Apéry numbers A(n) satisfy, 5, 73, 45,... η 7 (2τ)η 7 (3τ) η 5 (τ)η 5 = ( η 2 (τ)η 2 ) n (6τ) A(n) (6τ) η 2 (2τ)η 2. (3τ) n 0 FACT modular form O( 4 ) modular function O( 4 ) = e 2πiτ Not at all evidently, such a modular parametrization exists for all known Apéry-like numbers! 6 / 27

10 Apéry-like numbers and modular forms The Apéry numbers A(n) satisfy, 5, 73, 45,... η 7 (2τ)η 7 (3τ) η 5 (τ)η 5 = ( η 2 (τ)η 2 ) n (6τ) A(n) (6τ) η 2 (2τ)η 2. (3τ) n 0 FACT modular form O( 4 ) modular function O( 4 ) = e 2πiτ Not at all evidently, such a modular parametrization exists for all known Apéry-like numbers! As a conseuence, with z = 34x + x 2, Context: A(n)x n = n 0 f(τ) x(τ) y(x) 7 x z 4 2( + x + z) 3/2 3 F 2 ( 2, 2, 2, 024x ( x + z) 4 modular form of (integral) weight k modular function such that y(x(τ)) = f(τ) Then y(x) satisfies a linear differential euation of order k +. ). 6 / 27

11 Supercongruences for Apéry numbers Chowla, Cowles, Cowles (980) conjectured that, for primes p 5, A(p) 5 (mod p 3 ). 7 / 27

12 Supercongruences for Apéry numbers Chowla, Cowles, Cowles (980) conjectured that, for primes p 5, A(p) 5 (mod p 3 ). Gessel (982) proved that A(mp) A(m) (mod p 3 ). 7 / 27

13 Supercongruences for Apéry numbers Chowla, Cowles, Cowles (980) conjectured that, for primes p 5, A(p) 5 (mod p 3 ). Gessel (982) proved that A(mp) A(m) (mod p 3 ). THM Beukers, Coster 85, 88 The Apéry numbers satisfy the supercongruence (p 5) A(mp r ) A(mp r ) (mod p 3r ). 7 / 27

14 Supercongruences for Apéry numbers Chowla, Cowles, Cowles (980) conjectured that, for primes p 5, A(p) 5 (mod p 3 ). Gessel (982) proved that A(mp) A(m) (mod p 3 ). THM Beukers, Coster 85, 88 The Apéry numbers satisfy the supercongruence (p 5) A(mp r ) A(mp r ) (mod p 3r ). EG For primes p, simple combinatorics proves the congruence ( ) 2p = ( )( ) p p + (mod p 2 ). p k p k k For p 5, Wolstenholme s congruence shows that, in fact, ( ) 2p 2 (mod p 3 ). p 7 / 27

15 Conjecturally, supercongruences like A(mp r ) A(mp r ) (mod p 3r ) Robert Osburn (University of Dublin) Brundaban Sahu (NISER, India) hold for all Apéry-like numbers. Osburn Sahu 09 Current state of affairs for the six sporadic seuences from earlier: (a, b, c) (7, 5, ) (2, 4, 6) (0, 4, 64) (7, 3, 8) (, 5, 25) (9, 3, 27) A(n) ( n 2 ( n+k ) 2 k) Beukers, Coster k ( n ) 2 ( 2k k n k ( n k k n ) 2 Osburn Sahu S 4 ) 2 ( 2k )( 2(n k) ) k k ( )k 3 n 3k( n k ( )k( ) ( n 3 (4n 5k k k,l ( n k) 2 ( n l)( k l n k Osburn Sahu )( n+k ) (3k)! 3k n k! 3 open!! 3n )( k+l ) n ) + ( 4n 5k 3n modulo p 2 Amdeberhan 4 ) ) Osburn Sahu S 4 open 8 / 27

16 Non-super congruences are abundant a(mp r ) a(mp r ) (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 9 / 27

17 Non-super congruences are abundant a(mp r ) a(mp r ) (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 ±,..., x± d ] 9 / 27

18 Non-super congruences are abundant a(mp r ) a(mp r ) (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 ±,..., x± d ] ( ) a(n) n T n Z[[T ]]. n= This is a natural condition in formal group theory. If a() =, then (C) is euivalent to exp 9 / 27

19 Cooper s sporadic seuences Cooper s search for integral solutions to (n + ) 3 u n+ = (2n + )(an 2 + an + b)u n n(cn 2 + d)u n revealed three additional sporadic solutions: s 7 (n) = n k=0 k=0 ( n k ) 2 ( n + k k )( ) 2k n s 0 (n) = n s 0 and supercongruence known ) 4 [n/3] ( )( )( ) [( ) ( )] n 2k 2(n k) 2n 3k 2n 3k s 8 (n) = ( ) k + k k n k n n k=0 ( n k 0 / 27

20 Cooper s sporadic seuences Cooper s search for integral solutions to (n + ) 3 u n+ = (2n + )(an 2 + an + b)u n n(cn 2 + d)u n revealed three additional sporadic solutions: s 7 (n) = n k=0 k=0 ( n k ) 2 ( n + k k )( ) 2k n s 0 (n) = n s 0 and supercongruence known ) 4 [n/3] ( )( )( ) [( ) ( )] n 2k 2(n k) 2n 3k 2n 3k s 8 (n) = ( ) k + k k n k n n k=0 ( n k CONJ Cooper 202 s 7 (mp) s 7 (m) (mod p 3 ) p 3 s 8 (mp) s 8 (m) (mod p 2 ) 0 / 27

21 Cooper s sporadic seuences Cooper s search for integral solutions to (n + ) 3 u n+ = (2n + )(an 2 + an + b)u n n(cn 2 + d)u n revealed three additional sporadic solutions: s 7 (n) = n k=0 k=0 ( n k ) 2 ( n + k k )( ) 2k n s 0 (n) = n s 0 and supercongruence known ) 4 [n/3] ( )( )( ) [( ) ( )] n 2k 2(n k) 2n 3k 2n 3k s 8 (n) = ( ) k + k k n k n n k=0 ( n k CONJ Cooper 202 THM Osburn- Sahu-S 204 s 7 (mp) s 7 (m) (mod p 3 ) p 3 s 8 (mp) s 8 (m) (mod p 2 ) s 7 (mp r ) s 7 (mp r ) (mod p 3r ) p 5 s 8 (mp r ) s 8 (mp r ) (mod p 2r ) 0 / 27

22 Diagonals Given a series F (x,..., x d ) = n,...,n d 0 a(n,..., n d )x n xn d d, its diagonal coefficients are the coefficients a(n,..., n). EG x y has diagonal coefficients ( 2n n ). / 27

23 Diagonals Given a series F (x,..., x d ) = n,...,n d 0 a(n,..., n d )x n xn d d, its diagonal coefficients are the coefficients a(n,..., n). EG x y = (x + y) n n=0 has diagonal coefficients ( 2n n ). / 27

24 Diagonals Given a series F (x,..., x d ) = n,...,n d 0 a(n,..., n d )x n xn d d, its diagonal coefficients are the coefficients a(n,..., n). EG x y = (x + y) n n=0 has diagonal coefficients ( 2n n ). For comparison, their univariate generating function is n=0 ( ) 2n x n = n 4x. / 27

25 Diagonals Given a series F (x,..., x d ) = n,...,n d 0 a(n,..., n d )x n xn d d, its diagonal coefficients are the coefficients a(n,..., n). EG x y = (x + y) n n=0 has diagonal coefficients ( 2n n ). For comparison, their univariate generating function is n=0 ( ) 2n x n = n 4x. The diagonal of a rational function is D-finite. Gessel Zeilberger Lipshitz / 27

26 Apéry numbers as diagonals THM S 204 The Apéry numbers are the diagonal coefficients of ( x x 2 )( x 3 x 4 ) x x 2 x 3 x 4 2 / 27

27 Apéry numbers as diagonals THM S 204 The Apéry numbers are the diagonal coefficients of ( x x 2 )( x 3 x 4 ) x x 2 x 3 x 4 THM MacMahon 95 For x = (x,..., x n ) and m = (m,..., m n ) Z n 0, [x m n n ] det (I n BX) = [xm ] B i,j x j i= j= mi, where B C n n and X is the diagonal matrix with entries x,..., x n. 2 / 27

28 Apéry numbers as diagonals THM S 204 THM MacMahon 95 The Apéry numbers are the diagonal coefficients of = ( x x 2 )( x 3 x 4 ) x x 2 x 3 x 4 n Z 4 0 For x = (x,..., x n ) and m = (m,..., m n ) Z n 0, [x m n n ] det (I n BX) = [xm ] B i,j x j i= j= A(n)x n. mi, where B C n n and X is the diagonal matrix with entries x,..., x n. B = , X = x x 2 x 3 x 4 A(n) = [x n ](x + x 2 + x 3 ) n (x + x 2 ) n 2 (x 3 + x 4 ) n 3 (x 2 + x 3 + x 4 ) n 4 2 / 27

29 Apéry numbers as diagonals THM S 204 The Apéry numbers are the diagonal coefficients of = ( x x 2 )( x 3 x 4 ) x x 2 x 3 x 4 n Z 4 0 A(n)x n. The coefficients are the multivariate Apéry numbers A(n) = k Z ( n k )( n3 k )( )( n + n 2 k n3 + n 4 k n n 3 ). 2 / 27

30 Apéry numbers as diagonals THM S 204 The Apéry numbers are the diagonal coefficients of = ( x x 2 )( x 3 x 4 ) x x 2 x 3 x 4 n Z 4 0 A(n)x n. The coefficients are the multivariate Apéry numbers A(n) = k Z ( n k )( n3 k Univariate generating function: A(n)x n = n 0 where z = 34x + x 2. )( )( n + n 2 k n3 + n 4 k n 7 x z 4 2( + x + z) 3/2 3 F 2 ( 2, 2, 2, n 3 ). 024x ( x + z) 4 ), 2 / 27

31 Apéry numbers as diagonals THM S 204 The Apéry numbers are the diagonal coefficients of = ( x x 2 )( x 3 x 4 ) x 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 { (mod 8), if n even, A(n) 5 (mod 8), if n odd. Chowla Cowles Cowles 80 Rowland Yassawi 3 2 / 27

32 Multivariate supercongruences THM S 204 Define A(n) = A(n, n 2, n 3, n 4 ) by = ( x x 2 )( x 3 x 4 ) x 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 ) (mod p 3r ). 3 / 27

33 Multivariate supercongruences THM S 204 Define A(n) = A(n, n 2, n 3, n 4 ) by = ( x x 2 )( x 3 x 4 ) x 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 ) (mod p 3r ). a(n)x n = F (x) = a(pn)x pn = p n 0 n 0 p F (ζp k x) k=0 ζ p = e 2πi/p Hence, both A(np r ) and A(np r ) have rational generating function. The proof, however, relies on an explicit binomial sum for the coefficients. 3 / 27

34 Multivariate supercongruences THM S 204 Define A(n) = A(n, n 2, n 3, n 4 ) by = ( x x 2 )( x 3 x 4 ) x 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 ) (mod p 3r ). By MacMahon s Master Theorem, A(n) = ( )( )( )( ) n n3 n + n 2 k n3 + n 4 k. k k n n 3 k Z 3 / 27

35 Multivariate supercongruences THM S 204 Define A(n) = A(n, n 2, n 3, n 4 ) by = ( x x 2 )( x 3 x 4 ) x 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 ) (mod p 3r ). By MacMahon s Master Theorem, A(n) = ( )( )( )( ) n n3 n + n 2 k n3 + n 4 k. k k n n 3 k Z Because A(n ) = A( n, n, n, n), we also find A(mp r ) A(mp r ) (mod p 3r ). Beukers 85 3 / 27

36 Many more conjectural multivariate supercongruences Exhaustive search by Alin Bostan and Bruno Salvy: /( p(x, y, z, w)) with p(x, y, z, w) a sum of distinct monomials; Apéry numbers as diagonal (x + y + xy)(z + w + zw) ( + w)(z + xy + yz + zx + xyz) (y + z + xy + xz + zw + xyw + xyzw) (y + z + xz + wz + xyw + xzw + xyzw) (z + xy + yz + xw + xyw + yzw + xyzw) (z + (x + y)(z + w) + xyz + xyzw) 4 / 27

37 Many more conjectural multivariate supercongruences Exhaustive search by Alin Bostan and Bruno Salvy: /( p(x, y, z, w)) with p(x, y, z, w) a sum of distinct monomials; Apéry numbers as diagonal (x + y + xy)(z + w + zw) ( + w)(z + xy + yz + zx + xyz) (y + z + xy + xz + zw + xyw + xyzw) (y + z + xz + wz + xyw + xzw + xyzw) (z + xy + yz + xw + xyw + yzw + xyzw) (z + (x + y)(z + w) + xyz + xyzw) CONJ S 204 The coefficients B(n) of each of these satisfy, for p 5, B(np r ) B(np r ) (mod p 3r ). 4 / 27

38 An infinite family of rational functions THM S 204 Let λ Z l >0 with d = λ λ l. Define A λ (n) by j l [ r λ j x λ +...+λ j +r ] x x 2 x d = n Z d 0 A λ (n)x n. If l 2, then, for all primes p, A λ (np r ) A λ (np r ) (mod p 2r ). If l 2 and max(λ,..., λ l ) 2, then, for primes p 5, A λ (np r ) A λ (np r ) (mod p 3r ). EG λ = (2, 2) λ = (2, ) ( x x 2 )( x 3 x 4 ) x x 2 x 3 x 4 ( x x 2 )( x 3 ) x x 2 x 3 5 / 27

39 Further examples EG ( x x 2 )( x 3 ) x x 2 x 3 has as diagonal the Apéry-like numbers, associated with ζ(2), n ( ) n 2 ( ) n + k B(n) =. k k k=0 EG ( x )( x 2 ) ( x d ) x x 2 x d has as diagonal the numbers Y d (n) = n k=0 ( ) n d. k d = 3: Franel, d = 4: Yang Zudilin In each case, we obtain supercongruences generalizing results of Coster (988) and Chan Cooper Sica (200). 6 / 27

40 A conjectural multivariate supercongruence CONJ S 204 The coefficients Z(n) of = (x + x 2 + x 3 + x 4 ) + 27x x 2 x 3 x 4 n Z 4 0 satisfy, for p 5, the multivariate supercongruences Z(np r ) Z(np r ) (mod p 3r ). Z(n)x n Here, the diagonal coefficients are the Almkvist Zudilin numbers Z(n) = n ( )( ) n n + k (3k)! ( 3) n 3k 3k n k! 3, k=0 for which the univariate congruences are still open. 7 / 27

41 Basic -analogs The natural number n has the -analog: [n] = n = n In the limit a -analog reduces to the classical object. 8 / 27

42 Basic -analogs The natural number n has the -analog: [n] = n = n In the limit a -analog reduces to the classical object. The -factorial: The -binomial coefficient: ( ) n = k [n]! = [n] [n ] [] ( ) [n]! n [k]! [n k]! = n k 8 / 27

43 A -binomial coefficient EG ( ) 6 = = 3 5 ( ) 6 = ( )( ) / 27

44 A -binomial coefficient EG ( ) 6 = = 3 5 ( ) 6 = ( )( ) 2 + = ( + 2 ) ( ) ( ) }{{}}{{} =[3] =[5] 9 / 27

45 A -binomial coefficient EG ( ) 6 = = 3 5 ( ) 6 = ( )( ) 2 + = ( + 2 ) ( ) ( ) }{{}}{{}}{{} =Φ 6 () =[3] =[5] The cyclotomic polynomial Φ 6 () becomes for = and hence invisible in the classical world 9 / 27

46 The coefficients of -binomial coefficients Here s some -binomials in expanded form: EG ( ) 6 2 ( ) 9 3 = = The degree of the -binomial is k(n k). All coefficients are positive! In fact, the coefficients are unimodal. Sylvester, / 27

47 A few faces of the -binomial coefficient The -binomial coefficient ( ) n k satisfies a -version of Pascal s rule, ( ) n = j ( ) ( ) n n + j, j j 2 / 27

48 A few faces of the -binomial coefficient The -binomial coefficient ( ) n k satisfies a -version of Pascal s rule, ( ) n = j counts k-subsets of an n-set weighted by their sum, ( ) ( ) n n + j, j j 2 / 27

49 A few faces of the -binomial coefficient The -binomial coefficient ( ) n k satisfies a -version of Pascal s rule, ( ) n = j counts k-subsets of an n-set weighted by their sum, ( ) ( ) n n + j, j j features in a binomial theorem for noncommuting variables, (x + y) n = n j=0 ( ) n x j y n j, j if yx = xy, 2 / 27

50 A few faces of the -binomial coefficient The -binomial coefficient ( ) n k satisfies a -version of Pascal s rule, ( ) n = j counts k-subsets of an n-set weighted by their sum, ( ) ( ) n n + j, j j features in a binomial theorem for noncommuting variables, (x + y) n = n j=0 ( ) n x j y n j, j if yx = xy, has a -integral representation analogous to the beta function, 2 / 27

51 A few faces of the -binomial coefficient The -binomial coefficient ( ) n k satisfies a -version of Pascal s rule, ( ) n = j counts k-subsets of an n-set weighted by their sum, ( ) ( ) n n + j, j j features in a binomial theorem for noncommuting variables, (x + y) n = n j=0 ( ) n x j y n j, j if yx = xy, has a -integral representation analogous to the beta function, counts the number of k-dimensional subspaces of F n. 2 / 27

52 A -analog of Babbage s congruence Combinatorially, we again obtain: ( ) 2p = ( ) ( ) p p p k k p k (p k)2 -Chu-Vandermonde 22 / 27

53 A -analog of Babbage s congruence Combinatorially, we again obtain: ( ) 2p = ( ) ( ) p p p k k p k (p k)2 -Chu-Vandermonde p2 + = [2] p 2 (mod [p]2 ) ( ) p (Note that [p] divides unless k = 0 or k = p.) k 22 / 27

54 A -analog of Babbage s congruence Combinatorially, we again obtain: ( ) 2p = ( ) ( ) p p p k k p k (p k)2 -Chu-Vandermonde p2 + = [2] p 2 (mod [p]2 ) ( ) p (Note that [p] divides unless k = 0 or k = p.) k This combinatorial argument extends to show: ( ) ( ) THM ap a Clark (mod [p] 2 bp b ) 995 p2 22 / 27

55 A -analog of Babbage s congruence Combinatorially, we again obtain: ( ) 2p = ( ) ( ) p p p k k p k (p k)2 -Chu-Vandermonde p2 + = [2] p 2 (mod [p]2 ) ( ) p (Note that [p] divides unless k = 0 or k = p.) k This combinatorial argument extends to show: ( ) ( ) THM ap a Clark (mod [p] 2 bp b ) 995 p2 Similar results by Andrews; e.g.: ( ) ( ) ap a (a b)b(p 2) (mod [p] 2 bp b ) p 22 / 27

56 A -analog of Ljunggren s congruence The following answers the uestion of Andrews to find a -analog of Wolstenholme s congruence. THM S 20 For any prime p, ( ) ap bp ( ) a (a b)b b p2 ( ) a p 2 b 24 (p ) 2 (mod [p] 3 ). 23 / 27

57 A -analog of Ljunggren s congruence The following answers the uestion of Andrews to find a -analog of Wolstenholme s congruence. THM S 20 For any prime p, ( ) ap bp ( ) a (a b)b b p2 ( ) a p 2 b 24 (p ) 2 (mod [p] 3 ). EG Choosing p = 3, a = 2, and b =, we have ( ) 26 = ( 3 ) 2 + ( ) 3 f() 3 where f() = is an irreducible polynomial with integer coefficients. 23 / 27

58 A -analog of Ljunggren s congruence The following answers the uestion of Andrews to find a -analog of Wolstenholme s congruence. THM S 20 For any prime p, ( ) ap bp ( ) a (a b)b b p2 ( ) a p 2 b 24 (p ) 2 (mod [p] 3 ). Note that p2 24 is an integer if (p, 6) =. (The polynomial congruence holds for p = 2, 3 but coefficients are rational.) 23 / 27

59 A -analog of Ljunggren s congruence The following answers the uestion of Andrews to find a -analog of Wolstenholme s congruence. THM S 20 For any prime p, ( ) ap bp ( ) a (a b)b b p2 ( ) a p 2 b 24 (p ) 2 (mod [p] 3 ). Note that p2 24 is an integer if (p, 6) =. (The polynomial congruence holds for p = 2, 3 but coefficients are rational.) Ljunggren s classical congruence holds modulo p 3+r with r the p-adic valuation of ab(a b) ( a b). Jacobsthal 52 Is there a nice explanation or analog in the -world? 23 / 27

60 A -analog of Ljunggren s congruence The following answers the uestion of Andrews to find a -analog of Wolstenholme s congruence. THM S 20 For any prime p, ( ) ap bp ( ) a (a b)b b p2 ( ) a p 2 b 24 (p ) 2 (mod [p] 3 ). Note that p2 24 is an integer if (p, 6) =. (The polynomial congruence holds for p = 2, 3 but coefficients are rational.) Ljunggren s classical congruence holds modulo p 3+r with r the p-adic valuation of ab(a b) ( a b). Jacobsthal 52 Is there a nice explanation or analog in the -world? The congruence holds mod Φ n () 3 if p is replaced by any integer n. (No classical counterpart since Φ n () = unless n is a prime power.) 23 / 27

61 A -version of the Apéry numbers A symmetric -analog of the Apéry numbers: n ( ) n 2 ( n + k A (n) = (n k)2 k k k=0 Appear implicitly in work of Krattenthaler Rivoal Zudilin 06 ) 2 24 / 27

62 A -version of the Apéry numbers A symmetric -analog of the Apéry numbers: n ( ) n 2 ( n + k A (n) = (n k)2 k k k=0 Appear implicitly in work of Krattenthaler Rivoal Zudilin 06 The first few values are: A(0) = A (0) = A() = 5 A () = A(2) = 73 A (2) = A(3) = 445 A (3) = ) / 27

63 -supercongruences for the Apéry numbers THM S 205 The -Apéry numbers, defined as A (n) = n ( n (n k)2 k k=0 satisfy the supercongruences ) 2 ( n + k k ) 2, A (pn) A p 2 (n) p2 2 (p ) 2 f(n) (mod [p] 3 ). 25 / 27

64 -supercongruences for the Apéry numbers THM S 205 The -Apéry numbers, defined as A (n) = n ( n (n k)2 k k=0 satisfy the supercongruences ) 2 ( n + k k ) 2, A (pn) A p 2 (n) p2 2 (p ) 2 f(n) (mod [p] 3 ). The numbers f(n) can be expressed as 0, 5, 292, 3005, 52806,... f(n) = n ( ) n 2 ( ) n + k 2 k 4 g(n, k), g(n, k) = k(2n k) + k k (n + k) 2. k=0 Similar -analogs and congruences for other Apéry-like numbers? 25 / 27

65 Some of many open problems Supercongruences for all Apéry-like numbers proof of all the classical ones uniform explanation, proofs not relying on binomial sums Apéry-like numbers as diagonals find minimal rational functions extend supercongruences any structure? polynomial analogs of Apéry-like numbers find -analogs (e.g., for Almkvist Zudilin seuence) -supercongruences is there a geometric picture? Many further uestions remain. is the known list complete? higher-order analogs, Calabi Yau DEs modular supercongruences Beukers 87, Ahlgren Ono 00 ( ) p A a(p) (mod p 2 ), a(n) n = η 4 (2τ)η 4 (4τ) 2... n= 26 / 27

66 THANK YOU! Slides for this talk will be available from my website: A. Straub Multivariate Apéry numbers and supercongruences of rational functions Algebra & Number Theory, Vol. 8, Nr. 8, 204, p R. Osburn, B. Sahu, A. Straub Supercongruences for sporadic seuences to appear in Proceedings of the Edinburgh Mathematical Society, 204 A. Straub, W. Zudilin Positivity of rational functions and their diagonals to appear in Journal of Approximation Theory (special issue dedicated to Richard Askey), 204 M. Rogers, A. Straub A solution of Sun s $520 challenge concerning 520/π International Journal of Number Theory, Vol. 9, Nr. 5, 203, p J. Borwein, A. Straub, J. Wan, W. Zudilin (appendix by D. Zagier) Densities of short uniform random walks Canadian Journal of Mathematics, Vol. 64, Nr. 5, 202, p A. Straub A -analog of Ljunggren s binomial congruence DMTCS Proceedings: FPSAC 20, p / 27

67 Applications of Apéry-like numbers Random walks Jon Borwein Dirk Nuyens James Wan Wadim Zudilin U. of Newcastle, AU K.U.Leuven, BE SUTD, SG U. of Newcastle, AU Series for /π Mat Rogers U. of Montreal, CA Positivity of rational functions Wadim Zudilin U. of Newcastle, AU 28 / 38

68 Example I: Random walks n steps in the plane (length, random direction) 29 / 38

69 Example I: Random walks n steps in the plane (length, random direction) 29 / 38

70 Example I: Random walks n steps in the plane (length, random direction) 29 / 38

71 Example I: Random walks n steps in the plane (length, random direction) 29 / 38

72 Example I: Random walks n steps in the plane (length, random direction) 29 / 38

73 Example I: Random walks n steps in the plane (length, random direction) 29 / 38

74 Example I: Random walks n steps in the plane (length, random direction) d 29 / 38

75 Example I: Random walks n steps in the plane (length, random direction) p n (x) probability density of distance traveled p 3 (x) p 4 (x) d p 5 (x) p 6 (x) / 38

76 Example I: Random walks n steps in the plane (length, 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 () = 4 π, W 3() = 3 2 /3 6 π 4 Γ6 ( 3 ) /3 ( ) 2 4 π 4 Γ6 3 classical Borwein Nuyens S Wan, / 38

77 Example I: Random walks The probability moments W n (s) = include the Apéry-like numbers W 3 (2k) = W 4 (2k) = k j=0 k j=0 ( k j ( k j 0 ) 2 ( 2j j ) 2 ( 2j j x s p n (x) dx ), )( ) 2(k j). k j 30 / 38

78 Example I: Random walks The probability moments W n (s) = include the Apéry-like numbers W 3 (2k) = W 4 (2k) = k j=0 k j=0 ( k j ( k j 0 ) 2 ( 2j j ) 2 ( 2j j x s p n (x) dx ), )( ) 2(k j). k j THM Borwein- Nuyens- S-Wan 200 W n (2k) = a + +a n=k ( k a,..., a n ) 2 30 / 38

79 Example I: Random walks 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 p 4 (x) = 2 6 x 2 π 2 x p 5(0) = ( 3, 2 3 Re 3 F 2 ( 2, 2, 2 5 6, 7 6 x 2 ( 9 x 2) ) 2 (3 + x 2 ) 3 ( 6 x 2 ) ) 3 08x π 4 Γ( 5 )Γ( 2 5 )Γ( 4 5 )Γ( 8 5 ) easy classical with a spin new BSWZ 20 3 / 38

80 Example II: Series for /π 4 π = n=0 6 π = n=0 (/2) 3 n n! 3 (6n + ) 4 n (/2) 3 n n! 3 (42n + 5) 2 6n Srinivasa Ramanujan Modular euations and approximations to π Quart. J. Math., Vol. 45, p , / 38

81 Example II: Series for /π 4 π = n=0 6 π = n=0 π = (/2) 3 n n! 3 (6n + ) 4 n (/2) 3 n n! 3 (42n + 5) n=0 2 6n (4n)! n n! n Last series used by Gosper in 985 to compute 7, 526, 00 digits of π First proof of all of Ramanujan s 7 series by Borwein brothers Srinivasa Ramanujan Modular euations and approximations to π Quart. J. Math., Vol. 45, p , 94 Jonathan M. Borwein and Peter B. Borwein Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity Wiley, / 38

82 Example II: Series for /π Sato observed that series for π can be built from Apéry-like numbers: EG Chan- Chan-Liu 2003 For the Domb numbers D(n) = 8 3π = n=0 n k=0 ( n k ) 2 ( 2k k D(n) 5n + 2 6n. )( 2(n k) n k ), 33 / 38

83 Example II: Series for /π Sato observed that series for π can be built from Apéry-like numbers: EG Chan- Chan-Liu 2003 For the Domb numbers D(n) = 8 3π = n=0 n k=0 ( n k ) 2 ( 2k k D(n) 5n + 2 6n. )( 2(n k) n k ), Sun offered a $520 bounty for a proof the following series: THM Rogers-S π = n=0 054n n ( 2n n ) n k=0 ( n k ) 2 ( 2k n ) ( ) k 8 2k n 33 / 38

84 Example II: Series for /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 (/2) 3 n n! 3 (42n + 5) 6 = 26n π n=0 modular form a n (A + Bn)x(τ) n = Af(τ) + B x(τ) x (τ) f (τ) 34 / 38

85 Example II: Series for /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 (/2) 3 n n! 3 (42n + 5) 6 = 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 τ E2 (τ) = 6 πiτ. These are the main ingredients for series for /π. Mix and stir. 34 / 38

86 Example III: Positivity of rational functions A rational function F (x,..., x d ) = n,...,n d 0 is positive if a n,...,n d > 0 for all indices. a n,...,n d x n xn d d EG The following rational functions are positive. Szegő 33 Kaluza 33 S(x, y, z) = (x + y + z) (xy + yz + zx) A(x, y, z) = (x + y + z) + 4xyz Askey Gasper 72 S 08 Askey Gasper 77 Koornwinder 78 Ismail Tamhankar 79 Gillis Reznick Zeilberger 83 Both functions are on the boundary of positivity. 35 / 38

87 Example III: Positivity of rational functions A rational function F (x,..., x d ) = n,...,n d 0 is positive if a n,...,n d > 0 for all indices. a n,...,n d x n xn d d EG The following rational functions are positive. Szegő 33 Kaluza 33 S(x, y, z) = (x + y + z) (xy + yz + zx) A(x, y, z) = (x + y + z) + 4xyz Askey Gasper 72 S 08 Askey Gasper 77 Koornwinder 78 Ismail Tamhankar 79 Gillis Reznick Zeilberger 83 Both functions are on the boundary of positivity. The diagonal coefficients of A are the Franel numbers n k=0 ( ) n 3. k 35 / 38

88 Example III: Positivity of rational functions CONJ Kauers- Zeilberger 2008 The following rational function is positive: (x + y + z + w) + 2(yzw + xzw + xyw + xyz) + 4xyzw. Would imply conjectured positivity of Lewy Askey rational function (x + y + z + w) (xy + xz + xw + yz + yw + zw). Recent proof of non-negativity by Scott and Sokal, / 38

89 Example III: Positivity of rational functions CONJ Kauers- Zeilberger 2008 The following rational function is positive: (x + y + z + w) + 2(yzw + xzw + xyw + xyz) + 4xyzw. Would imply conjectured positivity of Lewy Askey rational function (x + y + z + w) (xy + xz + xw + yz + yw + zw). Recent proof of non-negativity by Scott and Sokal, 203 PROP S-Zudilin 203 The Kauers Zeilberger function has diagonal coefficients d n = n ( ) n 2 ( ) 2k 2. k n k=0 36 / 38

90 Positivity of rational functions Consider rational functions F = /p(x,..., x d ) with p a symmetric polynomial, linear in each variable. Q EG Under what condition(s) is the positivity of F implied by the positivity of its diagonal? is positive. (x + y) has positive diagonal but is not positive. + x + y 37 / 38

91 Positivity of rational functions Consider rational functions F = /p(x,..., x d ) with p a symmetric polynomial, linear in each variable. Q EG Under what condition(s) is the positivity of F implied by the positivity of its diagonal? is positive. (x + y) has positive diagonal but is not positive. + x + y is not positive. + x 37 / 38

92 Positivity of rational functions Consider rational functions F = /p(x,..., x d ) with p a symmetric polynomial, linear in each variable. Q EG Q Under what condition(s) is the positivity of F implied by the positivity of its diagonal? is positive. (x + y) has positive diagonal but is not positive. + x + y is not positive. + x F positive diagonal of F and F xd =0 are positive? 37 / 38

93 Positivity of rational functions Consider rational functions F = /p(x,..., x d ) with p a symmetric polynomial, linear in each variable. Q EG Q Under what condition(s) is the positivity of F implied by the positivity of its diagonal? is positive. (x + y) has positive diagonal but is not positive. + x + y is not positive. + x F positive diagonal of F and F xd =0 are positive? THM S-Zudilin 203 F (x, y) = + c (x + y) + c 2 xy diagonal of F and F y=0 are positive is positive 37 / 38

94 THANK YOU! Slides for this talk will be available from my website: A. Straub Multivariate Apéry numbers and supercongruences of rational functions Algebra & Number Theory, Vol. 8, Nr. 8, 204, p R. Osburn, B. Sahu, A. Straub Supercongruences for sporadic seuences to appear in Proceedings of the Edinburgh Mathematical Society, 204 A. Straub, W. Zudilin Positivity of rational functions and their diagonals to appear in Journal of Approximation Theory (special issue dedicated to Richard Askey), 204 M. Rogers, A. Straub A solution of Sun s $520 challenge concerning 520/π International Journal of Number Theory, Vol. 9, Nr. 5, 203, p J. Borwein, A. Straub, J. Wan, W. Zudilin (appendix by D. Zagier) Densities of short uniform random walks Canadian Journal of Mathematics, Vol. 64, Nr. 5, 202, p A. Straub 38 / 38