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Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT How far does a drunkard get? Graduate Student Colloquium Armin Straub Tulane University, New Orleans April 12, 2011 Joint with : Jon Borwein Dirk Nuyens James Wan

  1. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT The simple two-step case confirmed The average distance in two steps: � 1 � 1 � � e 2 πix + e 2 πiy � � d x d y = ? W 2 (1) = 0 0 Armin Straub How far does a drunkard get?

  2. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT The simple two-step case confirmed The average distance in two steps: � 1 � 1 � � e 2 πix + e 2 πiy � � d x d y = ? W 2 (1) = 0 0 Mathematica 7 and Maple 14 think the answer is 0. Armin Straub How far does a drunkard get?

  3. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT The simple two-step case confirmed The average distance in two steps: � 1 � 1 � e 2 πix + e 2 πiy � � � d x d y = ? W 2 (1) = 0 0 Mathematica 7 and Maple 14 think the answer is 0. There is always a 1-dimensional reduction: � 1 � d y = 4 � � 1 + e 2 πiy � W 2 (1) = π ≈ 1 . 27324 0 Armin Straub How far does a drunkard get?

  4. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT The simple two-step case confirmed The average distance in two steps: � 1 � 1 � � e 2 πix + e 2 πiy � � d x d y = ? W 2 (1) = 0 0 Mathematica 7 and Maple 14 think the answer is 0. There is always a 1-dimensional reduction: � 1 � d y = 4 � � 1 + e 2 πiy � W 2 (1) = π ≈ 1 . 27324 0 This is the average length of a random arc on a unit circle. Armin Straub How far does a drunkard get?

  5. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT The average distance for 3 and more steps � � � e 2 πix 1 + . . . + e 2 πix n � � s d x W n ( s ) := [0 , 1] n On a desktop: W 3 (1) ≈ 1 . 57459723755189365749 W 4 (1) ≈ 1 . 79909248 W 5 (1) ≈ 2 . 00816 In fact, W 5 (1) ≈ 2 . 0081618 was the best estimate we could compute directly, notwithstanding the availability of 256 cores at the Lawrence Berkeley National Laboratory. Armin Straub How far does a drunkard get?

  6. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT The average distance for 3 and more steps � � e 2 πix 1 + . . . + e 2 πix n � � � s d x W n ( s ) := [0 , 1] n On a desktop: W 3 (1) ≈ 1 . 57459723755189365749 W 4 (1) ≈ 1 . 79909248 W 5 (1) ≈ 2 . 00816 In fact, W 5 (1) ≈ 2 . 0081618 was the best estimate we could compute directly, notwithstanding the availability of 256 cores at the Lawrence Berkeley National Laboratory. Hard to evaluate numerically to high precision. For instance, Monte-Carlo integration gives approximations with an asymptotic √ error of O (1 / N ) where N is the number of sample points. Armin Straub How far does a drunkard get?

  7. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT The average distance for 3 and more steps � � e 2 πix 1 + . . . + e 2 πix n � � � s d x W n ( s ) := [0 , 1] n On a desktop: W 3 (1) ≈ 1 . 57459723755189365749 W 4 (1) ≈ 1 . 79909248 W 5 (1) ≈ 2 . 00816 In fact, W 5 (1) ≈ 2 . 0081618 was the best estimate we could compute directly, notwithstanding the availability of 256 cores at the Lawrence Berkeley National Laboratory. Hard to evaluate numerically to high precision. For instance, Monte-Carlo integration gives approximations with an asymptotic √ error of O (1 / N ) where N is the number of sample points. Closed forms as in the case n = 2 ? Armin Straub How far does a drunkard get?

  8. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT Can we guess W 3 (1) ? W 3 (1) = 1 . 57459723755189365749 . . . Armin Straub How far does a drunkard get?

  9. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT Can we guess W 3 (1) ? W 3 (1) = 1 . 57459723755189365749 . . . Idea If we suspect that a number x 0 can be written as x 0 = a 1 x 1 + . . . a n x n for other numbers x j and rational a j then this can be detected! Armin Straub How far does a drunkard get?

  10. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT Can we guess W 3 (1) ? W 3 (1) = 1 . 57459723755189365749 . . . Idea If we suspect that a number x 0 can be written as x 0 = a 1 x 1 + . . . a n x n for other numbers x j and rational a j then this can be detected! PSLQ takes numbers x = ( x 1 , x 2 , . . . , x n ) and tries to find integers m = ( m 1 , m 2 , . . . , m n ) , not all zero, such that x · m = m 1 x 1 + . . . + m n x n = 0 . The vector m is called an integer relation for x . In case that no relation is found, PSLQ provides a lower bound for the norm of any potential integer relation. Armin Straub How far does a drunkard get?

  11. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT Can we guess W 3 (1) ? Armin Straub How far does a drunkard get?

  12. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT Can we guess W 3 (1) ? Armin Straub How far does a drunkard get?

  13. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT Can we guess W 3 (1) ? Armin Straub How far does a drunkard get?

  14. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT Getting data: computing some moments The s th moment of the distance after n steps: � n � � s � � � e 2 πx k i W n ( s ) := d x � � � � [0 , 1] n k =1 n s = 1 s = 2 s = 3 s = 4 s = 5 s = 6 s = 7 2 1 . 273 2 . 000 3 . 395 6 . 000 10 . 87 20 . 00 37 . 25 3 1 . 575 3 . 000 6 . 452 15 . 00 36 . 71 93 . 00 241 . 5 4 1 . 799 4 . 000 10 . 12 28 . 00 82 . 65 256 . 0 822 . 3 5 2 . 008 5 . 000 14 . 29 45 . 00 152 . 3 545 . 0 2037 . 6 2 . 194 6 . 000 18 . 91 66 . 00 248 . 8 996 . 0 4186 . Armin Straub How far does a drunkard get?

  15. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT Getting data: computing some moments The s th moment of the distance after n steps: � n � � s � � � e 2 πx k i W n ( s ) := d x � � � � [0 , 1] n k =1 n s = 1 s = 2 s = 3 s = 4 s = 5 s = 6 s = 7 2 1 . 273 2 . 000 3 . 395 6 . 000 10 . 87 20 . 00 37 . 25 3 1 . 575 3 . 000 6 . 452 15 . 00 36 . 71 93 . 00 241 . 5 4 1 . 799 4 . 000 10 . 12 28 . 00 82 . 65 256 . 0 822 . 3 5 2 . 008 5 . 000 14 . 29 45 . 00 152 . 3 545 . 0 2037 . 6 2 . 194 6 . 000 18 . 91 66 . 00 248 . 8 996 . 0 4186 . W 2 (1) = 4 π Armin Straub How far does a drunkard get?

  16. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT Getting data: computing some moments The s th moment of the distance after n steps: � n � � s � � � e 2 πx k i W n ( s ) := d x � � � � [0 , 1] n k =1 n s = 1 s = 2 s = 3 s = 4 s = 5 s = 6 s = 7 2 1 . 273 2 . 000 3 . 395 6 . 000 10 . 87 20 . 00 37 . 25 3 1 . 575 3 . 000 6 . 452 15 . 00 36 . 71 93 . 00 241 . 5 4 1 . 799 4 . 000 10 . 12 28 . 00 82 . 65 256 . 0 822 . 3 5 2 . 008 5 . 000 14 . 29 45 . 00 152 . 3 545 . 0 2037 . 6 2 . 194 6 . 000 18 . 91 66 . 00 248 . 8 996 . 0 4186 . W 2 (1) = 4 W 3 (1) = 1 . 57459723755189 . . . = ? π Armin Straub How far does a drunkard get?

  17. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT Getting data: computing some moments The s th moment of the distance after n steps: � n � � s � � � e 2 πx k i W n ( s ) := d x � � � � [0 , 1] n k =1 n s = 1 s = 2 s = 3 s = 4 s = 5 s = 6 s = 7 2 1 . 273 2 . 000 3 . 395 6 . 000 10 . 87 20 . 00 37 . 25 3 1 . 575 3 . 000 6 . 452 15 . 00 36 . 71 93 . 00 241 . 5 4 1 . 799 4 . 000 10 . 12 28 . 00 82 . 65 256 . 0 822 . 3 5 2 . 008 5 . 000 14 . 29 45 . 00 152 . 3 545 . 0 2037 . 6 2 . 194 6 . 000 18 . 91 66 . 00 248 . 8 996 . 0 4186 . W 2 (1) = 4 W 3 (1) = 1 . 57459723755189 . . . = ? π Armin Straub How far does a drunkard get?

  18. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT Even moments n s = 2 s = 4 s = 6 s = 8 s = 10 Sloane’s 2 2 6 20 70 252 A000984 3 3 15 93 639 4653 A002893 4 4 28 256 2716 31504 A002895 5 5 45 545 7885 127905 6 6 66 996 18306 384156 Armin Straub How far does a drunkard get?

  19. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT Even moments n s = 2 s = 4 s = 6 s = 8 s = 10 Sloane’s 2 2 6 20 70 252 A000984 3 3 15 93 639 4653 A002893 4 4 28 256 2716 31504 A002895 5 5 45 545 7885 127905 6 6 66 996 18306 384156 Apparently: W n (2) = n Armin Straub How far does a drunkard get?

  20. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT Even moments n s = 2 s = 4 s = 6 s = 8 s = 10 Sloane’s 2 2 6 20 70 252 A000984 3 3 15 93 639 4653 A002893 4 4 28 256 2716 31504 A002895 5 5 45 545 7885 127905 6 6 66 996 18306 384156 Apparently: W n (2) = n Also: W n (10) ≡ n modulo 10 Armin Straub How far does a drunkard get?

  21. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT The integer sequence database Armin Straub How far does a drunkard get?

  22. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT The integer sequence database Armin Straub How far does a drunkard get?

  23. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT The integer sequence database Armin Straub How far does a drunkard get?

  24. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT A combinatorial formula for the even moments Theorem (Borwein-Nuyens-S-Wan) � � 2 k � W n (2 k ) = . a 1 , . . . , a n a 1 + ··· + a n = k Armin Straub How far does a drunkard get?

  25. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT A combinatorial formula for the even moments Theorem (Borwein-Nuyens-S-Wan) � � 2 k � W n (2 k ) = . a 1 , . . . , a n a 1 + ··· + a n = k f n ( k ) := W n (2 k ) counts the number of abelian squares : strings xy of length 2 k from an alphabet with n letters such that y is a permutation of x . Armin Straub How far does a drunkard get?

  26. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT A combinatorial formula for the even moments Theorem (Borwein-Nuyens-S-Wan) � � 2 k � W n (2 k ) = . a 1 , . . . , a n a 1 + ··· + a n = k f n ( k ) := W n (2 k ) counts the number of abelian squares : strings xy of length 2 k from an alphabet with n letters such that y is a permutation of x . Introduced by Erd˝ os and studied by others. Surely: f 1 ( k ) = 1 . Example acbc ccba is an abelian square. It contributes to f 3 (4) . Armin Straub How far does a drunkard get?

  27. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT A miracle? Example In the case of n = 2 we count abelian squares made from two letters. b a b a a a b a a b. Armin Straub How far does a drunkard get?

  28. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT A miracle? Example In the case of n = 2 we count abelian squares made from two letters. b a b a a a b a a b. � 2 k � It follows that f 2 ( k ) = . k Armin Straub How far does a drunkard get?

  29. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT A miracle? Example In the case of n = 2 we count abelian squares made from two letters. b a b a a a b a a b. � 2 k � It follows that f 2 ( k ) = . k Armin Straub How far does a drunkard get?

  30. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT A miracle? Example In the case of n = 2 we count abelian squares made from two letters. b a b a a a b a a b. � 2 k � Recall: � ∞ It follows that f 2 ( k ) = . x n e − x d x n ! = Γ( n + 1) = k 0 Γ( s + 1) = s Γ( s ) � 2 k � Γ(1 / 2) = √ π So: W 2 (2 k ) = k Armin Straub How far does a drunkard get?

  31. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT A miracle? Example In the case of n = 2 we count abelian squares made from two letters. b a b a a a b a a b. � 2 k � Recall: � ∞ It follows that f 2 ( k ) = . x n e − x d x n ! = Γ( n + 1) = k 0 Γ( s + 1) = s Γ( s ) � 2 k � Γ(1 / 2) = √ π So: W 2 (2 k ) = k � 1 � Putting k = 1 1! Γ 2 (3 / 2) = 4 1 2 we obtain = (1 / 2)! 2 = 1 / 2 π Armin Straub How far does a drunkard get?

  32. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT A miracle? Example In the case of n = 2 we count abelian squares made from two letters. b a b a a a b a a b. � 2 k � Recall: � ∞ It follows that f 2 ( k ) = . x n e − x d x n ! = Γ( n + 1) = k 0 Γ( s + 1) = s Γ( s ) � 2 k � Γ(1 / 2) = √ π So: W 2 (2 k ) = k � 1 � Putting k = 1 1! Γ 2 (3 / 2) = 4 1 2 we obtain = (1 / 2)! 2 = 1 / 2 π � s � Indeed: W 2 ( s ) = s/ 2 Armin Straub How far does a drunkard get?

  33. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT Other combinatorial consequences Convolutions: k � k � 2 � f n + m ( k ) = f n ( j ) f m ( k − j ) . j j =0 Armin Straub How far does a drunkard get?

  34. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT Other combinatorial consequences Convolutions: k � k � 2 � f n + m ( k ) = f n ( j ) f m ( k − j ) . j j =0 Recursions by Sister Celine, e.g.: ( k + 2) 2 f 3 ( k + 2) − (10 k 2 + 30 k + 23) f 3 ( k + 1) + 9( k + 1) 2 f 3 ( k ) = 0 . Armin Straub How far does a drunkard get?

  35. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT Functional equations For integers k � 0 , ( k + 2) 2 W 3 (2 k + 4) − (10 k 2 + 30 k + 23) W 3 (2 k + 2) + 9( k + 1) 2 W 3 (2 k ) = 0 . Armin Straub How far does a drunkard get?

  36. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT Functional equations For integers k � 0 , ( k + 2) 2 W 3 (2 k + 4) − (10 k 2 + 30 k + 23) W 3 (2 k + 2) + 9( k + 1) 2 W 3 (2 k ) = 0 . Theorem (Carlson) If f ( z ) is analytic for Re ( z ) � 0 , “nice”, and f (0) = 0 , f (1) = 0 , f (2) = 0 , . . . , then f ( z ) = 0 identically. Armin Straub How far does a drunkard get?

  37. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT Functional equations For integers k � 0 , ( k + 2) 2 W 3 (2 k + 4) − (10 k 2 + 30 k + 23) W 3 (2 k + 2) + 9( k + 1) 2 W 3 (2 k ) = 0 . Theorem (Carlson) If f ( z ) is analytic for Re ( z ) � 0 , “nice”, and f (0) = 0 , f (1) = 0 , f (2) = 0 , . . . , then f ( z ) = 0 identically. | f ( z ) | � Ae α | z | , and | f ( iy ) | � Be β | y | for β < π Armin Straub How far does a drunkard get?

  38. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT Functional equations For integers k � 0 , ( k + 2) 2 W 3 (2 k + 4) − (10 k 2 + 30 k + 23) W 3 (2 k + 2) + 9( k + 1) 2 W 3 (2 k ) = 0 . Theorem (Carlson) If f ( z ) is analytic for Re ( z ) � 0 , “nice”, and f (0) = 0 , f (1) = 0 , f (2) = 0 , . . . , then f ( z ) = 0 identically. W n ( s ) is nice! | f ( z ) | � Ae α | z | , and | f ( iy ) | � Be β | y | for β < π Armin Straub How far does a drunkard get?

  39. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT Functional Equations for W n ( s ) So we get complex functional equations like ( s +4) 2 W 3 ( s +4) − 2(5 s 2 +30 s +46) W 3 ( s +2)+9( s +2) 2 W 3 ( s ) = 0 . Armin Straub How far does a drunkard get?

  40. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT Functional Equations for W n ( s ) So we get complex functional equations like ( s +4) 2 W 3 ( s +4) − 2(5 s 2 +30 s +46) W 3 ( s +2)+9( s +2) 2 W 3 ( s ) = 0 . This gives analytic continuations of W n ( s ) to the complex plane, with poles at certain negative integers. 4 4 3 3 2 2 1 1 � 6 � 4 � 2 2 � 6 � 4 � 2 2 � 1 � 1 � 2 � 2 � 3 � 3 W 3 ( s ) W 4 ( s ) Armin Straub How far does a drunkard get?

  41. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT W 4 ( s ) in the complex plane Armin Straub How far does a drunkard get?

  42. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT W 4 ( s ) in the complex plane Armin Straub How far does a drunkard get?

  43. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT W 3 (1) = 1 . 57459723755189 . . . = ? � s � 2 k � � Easy: W 2 (2 k ) = . In fact, W 2 ( s ) = . k s/ 2 Armin Straub How far does a drunkard get?

  44. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT W 3 (1) = 1 . 57459723755189 . . . = ? � s � 2 k � � Easy: W 2 (2 k ) = . In fact, W 2 ( s ) = . k s/ 2 In the case n = 3 , k � k � 2 � 2 j � � W 3 (2 k ) = j j j =0 Armin Straub How far does a drunkard get?

  45. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT W 3 (1) = 1 . 57459723755189 . . . = ? � s � 2 k � � Easy: W 2 (2 k ) = . In fact, W 2 ( s ) = . k s/ 2 In the case n = 3 , k � k � 2 � 2 j � � W 3 (2 k ) = j j j =0 Idea: again, replace k by a complex variable Armin Straub How far does a drunkard get?

  46. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT Interlude: hypergeometric functions The hypergeometric function: � ∞ � a 1 , . . . , a p � x n ( a 1 ) n · · · ( a p ) n � � p F q � x = � b 1 , . . . , b q ( b 1 ) n · · · ( b q ) n n ! n =0 ( a ) n = a ( a + 1) · · · ( a + n − 1) is the Pochhammer symbol Armin Straub How far does a drunkard get?

  47. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT Interlude: hypergeometric functions The hypergeometric function: � ∞ � a 1 , . . . , a p � x n ( a 1 ) n · · · ( a p ) n � � p F q � x = � b 1 , . . . , b q ( b 1 ) n · · · ( b q ) n n ! n =0 ( a ) n = a ( a + 1) · · · ( a + n − 1) is the Pochhammer symbol Why hypergeometric ? ∞ c n where c n +1 � Geometric: = x c n n =0 Armin Straub How far does a drunkard get?

  48. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT Interlude: hypergeometric functions The hypergeometric function: � ∞ � a 1 , . . . , a p � x n ( a 1 ) n · · · ( a p ) n � � p F q � x = � b 1 , . . . , b q ( b 1 ) n · · · ( b q ) n n ! n =0 ( a ) n = a ( a + 1) · · · ( a + n − 1) is the Pochhammer symbol Why hypergeometric ? ∞ c n where c n +1 � Geometric: = x c n n =0 ∞ c n where c n +1 � Hypergeometric: = r ( n ) c n n =0 Armin Straub How far does a drunkard get?

  49. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT Interlude: hypergeometric functions The hypergeometric function: � ∞ � a 1 , . . . , a p � x n ( a 1 ) n · · · ( a p ) n � � p F q � x = � b 1 , . . . , b q ( b 1 ) n · · · ( b q ) n n ! n =0 ( a ) n = a ( a + 1) · · · ( a + n − 1) is the Pochhammer symbol Why hypergeometric ? ∞ c n where c n +1 � Geometric: = x c n n =0 ∞ c n where c n +1 � Hypergeometric: = r ( n ) c n n =0 r ( n ) = ( n + a 1 ) · · · ( n + a p ) x ( n + b 1 ) · · · ( n + b q ) n + 1 Armin Straub How far does a drunkard get?

  50. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT W 3 (1) = 1 . 57459723755189 . . . = ? � s � 2 k � � Easy: W 2 (2 k ) = . In fact, W 2 ( s ) = . k s/ 2 In the case n = 3 , k � k � 2 � 2 j � � W 3 (2 k ) = j j j =0 Armin Straub How far does a drunkard get?

  51. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT W 3 (1) = 1 . 57459723755189 . . . = ? � s � 2 k � � Easy: W 2 (2 k ) = . In fact, W 2 ( s ) = . k s/ 2 In the case n = 3 , � 1 k � k � 2 � 2 j � � � 2 , − k, − k � � W 3 (2 k ) = = 3 F 2 � 4 � j j 1 , 1 j =0 � �� � =: V 3 (2 k ) Armin Straub How far does a drunkard get?

  52. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT W 3 (1) = 1 . 57459723755189 . . . = ? � s � 2 k � � Easy: W 2 (2 k ) = . In fact, W 2 ( s ) = . k s/ 2 In the case n = 3 , � 1 k � k � 2 � 2 j � � � 2 , − k, − k � � W 3 (2 k ) = = 3 F 2 � 4 � j j 1 , 1 j =0 � �� � =: V 3 (2 k ) So by Carlson’s Theorem W 3 ( s ) = V 3 ( s ) , no!?!?? Armin Straub How far does a drunkard get?

  53. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT W 3 (1) = 1 . 57459723755189 . . . = ? Here’s Re ( W 3 ( s ) − V 3 ( s )) : 0.05 2 4 6 8 10 � 0.05 � 0.10 � 0.15 Armin Straub How far does a drunkard get?

  54. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT W 3 (1) = 1 . 57459723755189 . . . = ? Here’s Re ( W 3 ( s ) − V 3 ( s )) : 0.05 2 4 6 8 10 � 0.05 | V 3 ( − i ( s + 1)) / V 3 ( − is ) | : e π = 23 . 1407 . . . 23.0 � 0.10 22.8 22.6 � 0.15 22.4 20 40 60 80 100 120 140 Armin Straub How far does a drunkard get?

  55. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT W 3 (1) = 1 . 57459723755189 . . . = ? Here’s Re ( W 3 ( s ) − V 3 ( s )) : 0.05 2 4 6 8 10 � 0.05 | V 3 ( − i ( s + 1)) / V 3 ( − is ) | : e π = 23 . 1407 . . . 23.0 � 0.10 22.8 22.6 � 0.15 22.4 20 40 60 80 100 120 140 Armin Straub How far does a drunkard get?

  56. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT W 3 (1) = 1 . 57459723755189 . . . = ? Here’s Re ( W 3 ( s ) − V 3 ( s )) : 0.05 2 4 6 8 10 � 0.05 | V 3 ( − i ( s + 1)) / V 3 ( − is ) | : e π = 23 . 1407 . . . 23.0 � 0.10 22.8 22.6 � 0.15 22.4 Numerically: 20 40 60 80 100 120 140 V 3 (1) ≈ 1 . 574597 − . 126027 i Armin Straub How far does a drunkard get?

  57. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT W 3 (1) = 1 . 57459723755189 . . . = ? Theorem (Borwein-Nuyens-S-Wan) � 1 � � 2 , − k 2 , − k � 2 For integers k we have W 3 ( k ) = Re 3 F 2 � 4 . � 1 , 1 Armin Straub How far does a drunkard get?

  58. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT W 3 (1) = 1 . 57459723755189 . . . = ? Theorem (Borwein-Nuyens-S-Wan) � 1 � � 2 , − k 2 , − k � 2 For integers k we have W 3 ( k ) = Re 3 F 2 � 4 . � 1 , 1 Corollary (Borwein-Nuyens-S-Wan) � 1 � � 2 � 2 1 / 3 2 2 / 3 W 3 (1) = 3 + 27 π 4 Γ 6 π 4 Γ 6 16 3 4 3 Similar formulas for W 3 (3) , W 3 (5) , . . . Armin Straub How far does a drunkard get?

  59. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT Densities 0.5 0.7 0.8 0.4 0.6 0.6 0.5 0.3 0.4 0.4 0.3 0.2 p 2 ( x ) p 3 ( x ) p 4 ( x ) 0.2 0.2 0.1 0.1 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 2.5 3.0 1 2 3 4 0.35 0.30 0.35 0.30 0.30 0.25 0.25 0.25 0.20 0.20 0.20 0.15 0.15 0.15 p 5 ( x ) p 6 ( x ) 0.10 p 7 ( x ) 0.10 0.10 0.05 0.05 0.05 7 1 2 3 4 5 1 2 3 4 5 6 1 2 3 4 5 6 Armin Straub How far does a drunkard get?

  60. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT Densities 0.5 0.7 0.8 0.4 0.6 0.6 0.5 0.3 0.4 0.4 0.3 0.2 p 2 ( x ) p 3 ( x ) p 4 ( x ) 0.2 0.2 0.1 0.1 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 2.5 3.0 1 2 3 4 0.35 0.30 0.35 0.30 0.30 0.25 0.25 0.25 0.20 0.20 0.20 0.15 0.15 0.15 p 5 ( x ) p 6 ( x ) 0.10 p 7 ( x ) 0.10 0.10 0.05 0.05 0.05 7 1 2 3 4 5 1 2 3 4 5 6 1 2 3 4 5 6 p 4 and p 5 are C 0 p 6 and p 7 are C 1 p 2 n +4 , p 2 n +5 are C n Armin Straub How far does a drunkard get?

  61. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT Densities 0.5 0.7 0.8 0.4 0.6 0.6 0.5 0.3 0.4 0.4 0.3 0.2 p 2 ( x ) p 3 ( x ) p 4 ( x ) 0.2 0.2 0.1 0.1 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 2.5 3.0 1 2 3 4 0.35 0.30 0.35 0.30 0.30 0.25 0.25 0.25 0.20 0.20 0.20 0.15 0.15 0.15 p 5 ( x ) p 6 ( x ) 0.10 p 7 ( x ) 0.10 0.10 0.05 0.05 0.05 7 1 2 3 4 5 1 2 3 4 5 6 1 2 3 4 5 6 p 4 and p 5 are C 0 p 6 and p 7 are C 1 p 2 n +4 , p 2 n +5 are C n � ∞ xtJ 0 ( xt ) J n p n ( x ) = 0 ( t ) d t 0 Armin Straub How far does a drunkard get?

  62. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT Densities 0.5 0.7 0.8 0.4 0.6 0.6 0.5 0.3 0.4 0.4 0.3 0.2 p 2 ( x ) p 3 ( x ) p 4 ( x ) 0.2 0.2 0.1 0.1 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 2.5 3.0 1 2 3 4 0.35 0.30 0.35 0.30 0.004 0.30 0.25 0.25 0.25 0.20 0.20 0.20 0.002 0.15 0.15 0.15 p 5 ( x ) p 6 ( x ) 0.10 p 7 ( x ) 0.10 0.10 0.05 0.05 0.05 10 20 30 40 50 7 1 2 3 4 5 1 2 3 4 5 6 1 2 3 4 5 6 � 0.002 p 4 and p 5 are C 0 n = 4 , x = 2 � 0.004 p 6 and p 7 are C 1 p 2 n +4 , p 2 n +5 are C n � ∞ xtJ 0 ( xt ) J n p n ( x ) = 0 ( t ) d t 0 Armin Straub How far does a drunkard get?

  63. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT Hypergeometric formulae 0.5 0.7 0.8 0.4 0.6 0.6 0.5 0.3 0.4 0.4 0.3 0.2 p 2 ( x ) p 3 ( x ) p 4 ( x ) 0.2 0.2 0.1 0.1 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 2.5 3.0 1 2 3 4 2 p 2 ( x ) = √ easy 4 − x 2 π Armin Straub How far does a drunkard get?

  64. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT Hypergeometric formulae 0.5 0.7 0.8 0.4 0.6 0.6 0.5 0.3 0.4 0.4 0.3 0.2 p 2 ( x ) p 3 ( x ) p 4 ( x ) 0.2 0.2 0.1 0.1 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 2.5 3.0 1 2 3 4 2 p 2 ( x ) = √ easy 4 − x 2 π √ � � x 2 � 9 − x 2 � 2 � 1 3 , 2 p 3 ( x ) = 2 3 x � 3 (3 + x 2 ) 2 F 1 classical � (3 + x 2 ) 3 π 1 � with a spin Armin Straub How far does a drunkard get?

  65. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT Hypergeometric formulae 0.5 0.7 0.8 0.4 0.6 0.6 0.5 0.3 0.4 0.4 0.3 0.2 p 2 ( x ) p 3 ( x ) p 4 ( x ) 0.2 0.2 0.1 0.1 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 2.5 3.0 1 2 3 4 2 p 2 ( x ) = √ easy 4 − x 2 π √ � � x 2 � 9 − x 2 � 2 � 1 3 , 2 p 3 ( x ) = 2 3 x � 3 (3 + x 2 ) 2 F 1 classical � (3 + x 2 ) 3 π 1 � with a spin √ � 1 � � 16 − x 2 � 3 � 2 , 1 2 , 1 16 − x 2 p 4 ( x ) = 2 � 2 Re 3 F 2 new, BSWZ � π 2 5 6 , 7 108 x 4 x � 6 Armin Straub How far does a drunkard get?

  66. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT A straight line? 0.35 0.30 0.25 0.20 0.15 � ∞ xtJ 0 ( xt ) J 5 p 5 ( x ) = 0 ( t ) d t 0.10 0 0.05 1 2 3 4 5 Armin Straub How far does a drunkard get?

  67. Introduction Moments Combinatorics Consequences W 3 (1) Densities RMT A straight line? “the graphical construction, however carefully reinvestigated, did not permit of our consider- ing the curve to be anything but a straight 0.35 line. . . Even if it is not absolutely true, it exem- plifies the extraordinary power of such integrals 0.30 of J products to give extremely close approxima- 0.25 tions to such simple forms as horizontal lines.” 0.20 — Karl Pearson, 1906 0.15 � ∞ xtJ 0 ( xt ) J 5 p 5 ( x ) = 0 ( t ) d t 0.10 0 0.05 1 2 3 4 5 Armin Straub How far does a drunkard get?

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