On a q -analog of the Ap´ ery numbers International conference on orthogonal polynomials and q -series University of Central Florida celebrating Mourad E.H. Ismail Armin Straub May 12, 2015 University of Illinois at Urbana–Champaign n � n � 2 � n + k � 2 � A ( n ) = k k k =0 1 , 5 , 73 , 1445 , 33001 , 819005 , 21460825 , . . . On a q -analog of the Ap´ ery numbers Armin Straub 1 / 21
Positivity of rational functions All Taylor coefficients of the following function are positive: CONJ Kauers- Zeilberger 1 2008 1 − ( x + y + z + w ) + 2( yzw + xzw + xyw + xyz ) + 4 xyzw. • Among those present, Askey, Ismail, Koornwinder have contributed to understanding the positivity of (some) rational functions. On a q -analog of the Ap´ ery numbers Armin Straub 2 / 21
Positivity of rational functions All Taylor coefficients of the following function are positive: CONJ Kauers- Zeilberger 1 2008 1 − ( x + y + z + w ) + 2( yzw + xzw + xyw + xyz ) + 4 xyzw. • Among those present, Askey, Ismail, Koornwinder have contributed to understanding the positivity of (some) rational functions. The diagonal coefficients of the Kauers–Zeilberger function are PROP S-Zudilin 2015 n � n � 2 � 2 k � 2 � D ( n ) = . k n k =0 • D ( n ) is an example of an Ap´ ery-like sequence . On a q -analog of the Ap´ ery numbers Armin Straub 2 / 21
Positivity of rational functions All Taylor coefficients of the following function are positive: CONJ Kauers- Zeilberger 1 2008 1 − ( x + y + z + w ) + 2( yzw + xzw + xyw + xyz ) + 4 xyzw. • Among those present, Askey, Ismail, Koornwinder have contributed to understanding the positivity of (some) rational functions. The diagonal coefficients of the Kauers–Zeilberger function are PROP S-Zudilin 2015 n � n � 2 � 2 k � 2 � D ( n ) = . k n k =0 • D ( n ) is an example of an Ap´ ery-like sequence . Can we conclude the conjectured positivity from the positivity of Q S-Zudilin 1 D ( n ) together with the (obvious) positivity of 1 − ( x + y + z )+2 xyz ? 2015 On a q -analog of the Ap´ ery numbers Armin Straub 2 / 21
Ap´ ery numbers and the irrationality of ζ (3) • The Ap´ ery numbers 1 , 5 , 73 , 1445 , . . . n � n � 2 � n + k � 2 � A ( n ) = k k satisfy k =0 ( n + 1) 3 A ( n + 1) = (2 n + 1)(17 n 2 + 17 n + 5) A ( n ) − n 3 A ( n − 1) . On a q -analog of the Ap´ ery numbers Armin Straub 3 / 21
Ap´ ery numbers and the irrationality of ζ (3) • The Ap´ ery numbers 1 , 5 , 73 , 1445 , . . . n � n � 2 � n + k � 2 � A ( n ) = k k satisfy k =0 ( n + 1) 3 A ( n + 1) = (2 n + 1)(17 n 2 + 17 n + 5) A ( n ) − n 3 A ( 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 � 2 � n + k � 2 n k ( − 1) m − 1 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. On a q -analog of the Ap´ ery numbers Armin Straub 3 / 21
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 . Q Are there other tuples ( a, b, c ) for which the solution defined by Beukers, u − 1 = 0 , u 0 = 1 is integral? Zagier On a q -analog of the Ap´ ery numbers Armin Straub 4 / 21
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 . Q Are there other tuples ( a, b, c ) for which the solution defined by 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 (with generating functions � 1 � � � � α, 1 − α � 2 2 , α, 1 − α 1 − C α z � � 3 F 2 � 4 C α z , 1 − C α z 2 F 1 , � � 1 , 1 1 1 − C α z � 6 and C α = 2 4 , 3 3 , 2 6 , 2 4 · 3 3 ) with α = 1 2 , 1 3 , 1 4 , 1 • 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) On a q -analog of the Ap´ ery numbers Armin Straub 4 / 21
The six sporadic Ap´ ery-like numbers ( a, b, c ) A ( n ) � n � 2 � n + k � 2 � (17 , 5 , 1) Ap´ ery numbers k n k � n � 2 � 2 k � 2 � (12 , 4 , 16) k n k � n � 2 � 2 k �� 2( n − k ) � � (10 , 4 , 64) Domb numbers n − k k k k � n �� n + k � (3 k )! � ( − 1) k 3 n − 3 k (7 , 3 , 81) Almkvist–Zudilin numbers k ! 3 3 k n k � n � 3 �� 4 n − 5 k − 1 � � 4 n − 5 k �� � ( − 1) k (11 , 5 , 125) + 3 n 3 n k k � n � 2 � n �� k �� k + l � � (9 , 3 , − 27) k l l n k,l On a q -analog of the Ap´ ery numbers Armin Straub 5 / 21
Ap´ ery-like numbers and modular forms • The Ap´ ery numbers A ( n ) satisfy 1 , 5 , 73 , 1145 , . . . � η 12 ( τ ) η 12 (6 τ ) � n η 7 (2 τ ) η 7 (3 τ ) � = A ( n ) . η 5 ( τ ) η 5 (6 τ ) η 12 (2 τ ) η 12 (3 τ ) n � 0 modular form modular function 1 + 5 q + 13 q 2 + 23 q 3 + O ( q 4 ) q − 12 q 2 + 66 q 3 + O ( q 4 ) q = e 2 πiτ On a q -analog of the Ap´ ery numbers Armin Straub 6 / 21
Ap´ ery-like numbers and modular forms • The Ap´ ery numbers A ( n ) satisfy 1 , 5 , 73 , 1145 , . . . � η 12 ( τ ) η 12 (6 τ ) � n η 7 (2 τ ) η 7 (3 τ ) � = A ( n ) . η 5 ( τ ) η 5 (6 τ ) η 12 (2 τ ) η 12 (3 τ ) n � 0 modular form modular function 1 + 5 q + 13 q 2 + 23 q 3 + O ( q 4 ) q − 12 q 2 + 66 q 3 + O ( q 4 ) q = e 2 πiτ Not at all evidently, such a modular parametrization exists for FACT all known Ap´ ery-like numbers! On a q -analog of the Ap´ ery numbers Armin Straub 6 / 21
Ap´ ery-like numbers and modular forms • The Ap´ ery numbers A ( n ) satisfy 1 , 5 , 73 , 1145 , . . . � η 12 ( τ ) η 12 (6 τ ) � n η 7 (2 τ ) η 7 (3 τ ) � = A ( n ) . η 5 ( τ ) η 5 (6 τ ) η 12 (2 τ ) η 12 (3 τ ) n � 0 modular form modular function 1 + 5 q + 13 q 2 + 23 q 3 + O ( q 4 ) q − 12 q 2 + 66 q 3 + O ( q 4 ) q = e 2 πiτ Not at all evidently, such a modular parametrization exists for FACT all known Ap´ ery-like numbers! √ • As a consequence, with z = 1 − 34 x + x 2 , � 1 � � 2 , 1 2 , 1 17 − x − z 1024 x � A ( n ) x n = � 2 √ 2(1 + x + z ) 3 / 2 3 F 2 � − . � (1 − x + z ) 4 1 , 1 4 n � 0 • Context: f ( τ ) modular form of (integral) 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 . On a q -analog of the Ap´ ery numbers Armin Straub 6 / 21
Supercongruences for Ap´ ery numbers • Chowla, Cowles, Cowles (1980) conjectured that, for primes p � 5 , (mod p 3 ) . A ( p ) ≡ 5 On a q -analog of the Ap´ ery numbers Armin Straub 7 / 21
Supercongruences for Ap´ ery numbers • Chowla, Cowles, Cowles (1980) conjectured that, for primes p � 5 , (mod p 3 ) . A ( p ) ≡ 5 (mod p 3 ) . • Gessel (1982) proved that A ( mp ) ≡ A ( m ) On a q -analog of the Ap´ ery numbers Armin Straub 7 / 21
Supercongruences for Ap´ ery numbers • Chowla, Cowles, Cowles (1980) conjectured that, for primes 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 On a q -analog of the Ap´ ery numbers Armin Straub 7 / 21
Supercongruences for Ap´ ery numbers • Chowla, Cowles, Cowles (1980) conjectured that, for primes 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 EG For primes p , simple combinatorics proves the congruence � 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 On a q -analog of the Ap´ ery numbers Armin Straub 7 / 21
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 ) � n � 2 � n + k � 2 � (17 , 5 , 1) Beukers, Coster ’87-’88 k k n � n � 2 � 2 k � 2 � (12 , 4 , 16) Osburn–Sahu–S ’14 k k n � n � 2 � 2 k �� 2( n − k ) � � (10 , 4 , 64) Osburn–Sahu ’11 k k k n − k 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 �� � k ( − 1) k � n � � 4 n − 5 k (11 , 5 , 125) + Osburn–Sahu–S ’14 k 3 n 3 n � � n � 2 � n �� k �� k + l � (9 , 3 , − 27) open k,l k l l n On a q -analog of the Ap´ ery numbers Armin Straub 8 / 21
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