Ap´ ery numbers and their experimental siblings Challenges in 21st Century Experimental Mathematical Computation ICERM, Brown University Armin Straub July 22, 2014 University of Illinois at Urbana–Champaign n � 2 � n + k � 2 � n � A ( n ) = k k k =0 1 , 5 , 73 , 1445 , 33001 , 819005 , 21460825 , . . . Jon Borwein Dirk Nuyens James Wan Wadim Zudilin Robert Osburn Brundaban Sahu Mathew Rogers Ap´ ery numbers and their experimental siblings Armin Straub 1 / 20
Ap´ ery numbers and the irrationality of ζ (3) • The Ap´ ery numbers 1 , 5 , 73 , 1445 , . . . n � 2 � n + k � 2 � n � A ( n ) = k k satisfy k =0 ( n + 1) 3 u n +1 = (2 n + 1)(17 n 2 + 17 n + 5) u n − n 3 u n − 1 . Ap´ ery numbers and their experimental siblings Armin Straub 2 / 20
Ap´ ery numbers and the irrationality of ζ (3) • The Ap´ ery numbers 1 , 5 , 73 , 1445 , . . . n � 2 � n + k � 2 � n � A ( n ) = k k satisfy k =0 ( n + 1) 3 u n +1 = (2 n + 1)(17 n 2 + 17 n + 5) u n − n 3 u n − 1 . ζ (3) = � ∞ 1 THM n 3 is irrational. n =1 Ap´ ery ’78 The same recurrence is satisfied by the “near”-integers proof n n k � 2 � n + k � 2 ( − 1) m − 1 � n 1 � � � . B ( n ) = j 3 + 2 m 3 � n �� n + m � k k m m j =1 m =1 k =0 Then, B ( n ) A ( n ) → ζ (3) . But too fast for ζ (3) to be rational. Ap´ ery numbers and their experimental siblings Armin Straub 2 / 20
Zagier’s search and Ap´ ery-like numbers • Recurrence for Ap´ ery numbers is the case ( a, b, c ) = (17 , 5 , 1) of ( n + 1) 3 u n +1 = (2 n + 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 Q Beukers, u − 1 = 0 , u 0 = 1 is integral? Zagier Ap´ ery numbers and their experimental siblings Armin Straub 3 / 20
Zagier’s search and Ap´ ery-like numbers • Recurrence for Ap´ ery numbers is the case ( a, b, c ) = (17 , 5 , 1) of ( n + 1) 3 u n +1 = (2 n + 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 Q Beukers, u − 1 = 0 , u 0 = 1 is integral? Zagier • Essentially, only 14 tuples ( a, b, c ) found. (Almkvist–Zudilin) • 4 hypergeometric and 4 Legendrian solutions • 6 sporadic solutions • Similar (and intertwined) story for: • ( n + 1) 2 u n +1 = ( an 2 + an + b ) u n − cn 2 u n − 1 (Beukers, Zagier) • ( n + 1) 3 u n +1 = (2 n + 1)( an 2 + an + b ) u n − n ( cn 2 + d ) u n − 1 (Cooper) Ap´ ery numbers and their experimental siblings Armin Straub 3 / 20
Ap´ ery-like numbers • Hypergeometric and Legendrian solutions have generating functions � 1 � 2 � � 2 , α, 1 − α � 1 � α, 1 − α − C α z � � 3 F 2 � 4 C α z , 1 − C α z 2 F 1 , � � 1 , 1 1 1 − C α z � with α = 1 2 , 1 3 , 1 4 , 1 6 and C α = 2 4 , 3 3 , 2 6 , 2 4 · 3 3 . • The six sporadic solutions are: ( a, b, c ) A ( n ) � (3 k )! k ( − 1) k 3 n − 3 k � n �� n + k (7 , 3 , 81) � 3 k n k ! 3 � 3 �� 4 n − 5 k − 1 �� k ( − 1) k � n � 4 n − 5 k � � (11 , 5 , 125) + k 3 n 3 n � 2 � 2 k � n �� 2( n − k ) � (10 , 4 , 64) � k k k n − k � 2 � 2 � 2 k � n � (12 , 4 , 16) k k n � 2 � n �� k + l � n �� k � � (9 , 3 , − 27) k,l k l l n � 2 � 2 � n + k � n (17 , 5 , 1) � k k n Ap´ ery numbers and their experimental siblings Armin Straub 4 / 20
Modularity of Ap´ ery-like numbers • The Ap´ ery numbers 1 , 5 , 73 , 1145 , . . . n � 2 � n + k � 2 � n � A ( n ) = k k satisfy k =0 � η ( τ ) η (6 τ ) � 12 n η 7 (2 τ ) η 7 (3 τ ) � = A ( n ) . η 5 ( τ ) η 5 (6 τ ) η (2 τ ) η (3 τ ) n � 0 modular form modular function Ap´ ery numbers and their experimental siblings Armin Straub 5 / 20
Modularity of Ap´ ery-like numbers • The Ap´ ery numbers 1 , 5 , 73 , 1145 , . . . n � 2 � n + k � 2 � n � A ( n ) = k k satisfy k =0 � η ( τ ) η (6 τ ) � 12 n η 7 (2 τ ) η 7 (3 τ ) � = A ( n ) . η 5 ( τ ) η 5 (6 τ ) η (2 τ ) η (3 τ ) n � 0 modular form modular function Not at all evidently, such a modular parametrization exists for FACT all known Ap´ ery-like numbers! • Context: f ( τ ) modular form of weight k x ( τ ) modular function y ( x ) such that y ( x ( τ )) = f ( τ ) Then y ( x ) satisfies a linear differential equation of order k + 1 . Ap´ ery numbers and their experimental siblings Armin Straub 5 / 20
Supercongruences for Ap´ ery numbers • Chowla, Cowles and Cowles (1980) conjectured that, for p � 5 , mod p 3 . A ( p ) ≡ 5 Ap´ ery numbers and their experimental siblings Armin Straub 6 / 20
Supercongruences for Ap´ ery numbers • Chowla, Cowles and Cowles (1980) conjectured that, for p � 5 , mod p 3 . A ( p ) ≡ 5 mod p 3 . • Gessel (1982) proved that A ( mp ) ≡ A ( m ) Ap´ ery numbers and their experimental siblings Armin Straub 6 / 20
Supercongruences for Ap´ ery numbers • Chowla, Cowles and Cowles (1980) conjectured that, for p � 5 , mod p 3 . A ( p ) ≡ 5 mod p 3 . • Gessel (1982) proved that A ( mp ) ≡ A ( m ) The Ap´ ery numbers satisfy the supercongruence ( p � 5 ) THM Beukers, Coster A ( mp r ) ≡ A ( mp r − 1 ) mod p 3 r . ’85, ’88 Ap´ ery numbers and their experimental siblings Armin Straub 6 / 20
Supercongruences for Ap´ ery numbers • Chowla, Cowles and Cowles (1980) conjectured that, for p � 5 , mod p 3 . A ( p ) ≡ 5 mod p 3 . • Gessel (1982) proved that A ( mp ) ≡ A ( m ) The Ap´ ery numbers satisfy the supercongruence ( p � 5 ) THM Beukers, Coster A ( mp r ) ≡ A ( mp r − 1 ) mod p 3 r . ’85, ’88 n � 2 � n + k � 2 EG � n � Mathematica 7 miscomputes A ( n ) = for n > 5500 . k k k =0 A (5 · 11 3 ) = 12488301 . . . about 2000 digits . . . about 8000 digits . . . 795652125 Weirdly, with this wrong value, one still has A (5 · 11 3 ) ≡ A (5 · 11 2 ) mod 11 6 . Ap´ ery numbers and their experimental siblings Armin Straub 6 / 20
Supercongruences for Ap´ ery numbers • Chowla, Cowles and Cowles (1980) conjectured that, for p � 5 , mod p 3 . A ( p ) ≡ 5 mod p 3 . • Gessel (1982) proved that A ( mp ) ≡ A ( m ) The Ap´ ery numbers satisfy the supercongruence ( p � 5 ) THM Beukers, Coster A ( mp r ) ≡ A ( mp r − 1 ) mod p 3 r . ’85, ’88 Simple combinatorics proves the congruence EG � 2 p � � p �� � p � mod p 2 . = ≡ 1 + 1 p k p − k k For p � 5 , Wolstenholme’s congruence shows that, in fact, � 2 p � mod p 3 . ≡ 2 p Ap´ ery numbers and their experimental siblings Armin Straub 6 / 20
Supercongruences for Ap´ ery-like numbers • Conjecturally, supercongruences like A ( mp r ) ≡ A ( mp r − 1 ) mod p 3 r Robert Osburn Brundaban Sahu (University of Dublin) (NISER, India) hold for all Ap´ ery-like numbers. Osburn–Sahu ’09 • Current state of affairs for the six sporadic sequences from earlier: ( a, b, c ) A ( n ) k ( − 1) k 3 n − 3 k � n � (3 k )! �� n + k (7 , 3 , 81) � modulo p 2 open!! 3 k n k ! 3 Amdeberhan ’14 � 3 �� 4 n − 5 k − 1 �� � 4 n − 5 k k ( − 1) k � n � (11 , 5 , 125) � + Osburn–Sahu–S ’14 k 3 n 3 n � 2 � 2 k �� 2( n − k ) � n � (10 , 4 , 64) � Osburn–Sahu ’11 k k k n − k � 2 � 2 k � 2 � n (12 , 4 , 16) � Osburn–Sahu–S ’14 k k n � 2 � n � n �� k �� k + l � � (9 , 3 , − 27) open k,l k l l n � 2 � 2 � n + k � n � (17 , 5 , 1) Beukers, Coster ’87-’88 k k n Ap´ ery numbers and their experimental siblings Armin Straub 7 / 20
A generalization: multivariate supercongruences THM Define A ( n ) = A ( n 1 , n 2 , n 3 , n 4 ) by S 2013 1 � = A ( n ) x n . (1 − x 1 − x 2 )(1 − x 3 − x 4 ) − x 1 x 2 x 3 x 4 n ∈ Z 4 � 0 • The Ap´ ery numbers are the diagonal coefficients . • For p � 5 , we have the multivariate supercongruences A ( n p r ) ≡ A ( n p r − 1 ) mod p 3 r . Ap´ ery numbers and their experimental siblings Armin Straub 8 / 20
A generalization: multivariate supercongruences THM Define A ( n ) = A ( n 1 , n 2 , n 3 , n 4 ) by S 2013 1 � = A ( n ) x n . (1 − x 1 − x 2 )(1 − x 3 − x 4 ) − x 1 x 2 x 3 x 4 n ∈ Z 4 � 0 • The Ap´ ery numbers are the diagonal coefficients . • For p � 5 , we have the multivariate supercongruences A ( n p r ) ≡ A ( n p r − 1 ) mod p 3 r . • Both A ( n p r ) and A ( n p r − 1 ) have rational generating function. The proof, however, relies on an explicit binomial sum for the coefficients. Ap´ ery numbers and their experimental siblings Armin Straub 8 / 20
Short random walks joint work with: Jon Borwein Dirk Nuyens James Wan Wadim Zudilin U. Newcastle, AU K.U.Leuven, BE SUTD, SG U. Newcastle, AU Ap´ ery numbers and their experimental siblings Armin Straub 9 / 20
Random walks in the plane n steps in the plane (length 1 , random direction) Ap´ ery numbers and their experimental siblings Armin Straub 10 / 20
Random walks in the plane n steps in the plane (length 1 , random direction) Ap´ ery numbers and their experimental siblings Armin Straub 10 / 20
Random walks in the plane n steps in the plane (length 1 , random direction) Ap´ ery numbers and their experimental siblings Armin Straub 10 / 20
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