The Fermat cubic, elliptic functions, continued fractions, and a - PowerPoint PPT Presentation

2 y 1 0 -1 0 1 2 x -1 The Fermat cubic, elliptic functions, continued fractions, and a combinatorial excursion Philippe Flajolet INRIA Rocq. (France) Grenoble, S´ eorie des Nombres — February 14, 2007 eminaire de Th´ Based on joint work with Eric Conrad , Columbus, OH (USA). In S´ eminaire Lotharingien de Combinatoire vol 54, 44 pages Feb 2005: A LGO Sem. Rocquencourt; Apr 2005: SLC54, Lucelle. Feb 2007: UJF , Grenoble. 1

1 “ALGEBRAIC” CONTINUED FRACTIONS S (Stieltjes) J (Jacobi) 1 1 1 n ! · z n = � z tan z = , . 1 2 · z 2 z 2 n ≥ 0 1 − 1 − 1 · z − 2 2 · z 2 z 2 3 − 1 − 3 · z − ... z 2 5 − ... CF: Iterate X �→ 1 /X ; X = ⌊ X ⌋ + { X } . Here: f = f (0) + zf ′ (0) + z 2 “ { f } ”. (Irrationality of π (Lambert) and summation of divergent series (Euler). Also re- lated to orthogonal polynomials Pad´ e approximants, moment problems, etc.) Explicit CFs are very rare : From Perron, Wall, Chihara, etc, perhaps less than 100 continued fractions are known for special functions. 2

ζ (3) = � 1 /n 3 is irrational. Theorem (Ap´ ery 1978): 6 ζ (3) = , with ̟ ( n ) := (2 n + 1)(17 n ( n + 1) + 5) . 1 6 ̟ (0) − 2 6 ̟ (1) − 3 6 ̟ (2) − ... 1 1 � (Stieltjes) ( n + z ) 3 = , 1 6 n ≥ 0 σ (0) − 2 6 σ (1) − 3 6 σ (2) − ... with Cf Berndt/Ramanujan. σ ( n ) = (2 n + 1)(2 z ( z + 1) + n ( n + 1) + 1) . 3

Theorem (Conrad 2002): For a certain function sm : Z ∞ x 2 sm( u ) e − u/x du = , 1 · 2 2 · 3 2 · 4 x 6 0 1 + b 0 x 3 − 4 · 5 2 · 6 2 · 7 x 6 1 + b 1 x 3 − 7 · 8 2 · 9 2 · 10 x 6 1 + b 2 x 3 − ... b n = 2(3 n + 1)((3 n + 1) 2 + 1) , where and Z z » 1 – dt 3 , 2 3 , 4 3; z 3 sm( z ) = Inv (1 − t 3 ) 2 / 3 = Inv z · 2 F 1 . 0 4

Plan: some cute combinatorics surrounding the functions — The Fermat cubic x 3 + y 3 = 1 and Dixonian functions — A first model related to P´ olya urns and branching processes — A second model of Dixonian function by permutations ⋆ based on parity constraints [cf Viennot, F ., Dumont] — A third model of Dixonian function by weighted Dyck paths, related to continued fractions, and permutations ⋆ based on patterns of order 3 [cf F .-Franc ¸ on] Side effects: An analytic-combinatorial approach to urn processes that are 2 × 2 balanced. 5

2 FERMAT CURVES: CIRCLE & CUBIC The Fermat curve F m is the complex algebraic curve x m + y m = 1 . Circle F 2 : Consider s ′ = c, c ′ = − s , with s (0) = 0 , c (0) = 1 . The transcendental functions s, c do parameterize the circle, s ( z ) 2 + c ( z ) 2 = 1 , ( s 2 + c 2 ) ′ = 2 ss ′ + 2 cc ′ = 2 sc − 2 cs = 0 . since R Also: inversion from abelian integral R ( z, y ) dz on F 2 : Z sin z dt p (1 − t 2 ) 1 / 2 = z, cos( z ) = 1 − sin( z ) 2 0 sin z 1 For combinatorialists: tan z = sec z = cos z enumerate alternating cos z , (aka up-and-down, zig-zag) permutations [D´ esir´ e Andr´ e, 1881]. 6

The “complexity” of integral calculus over an algebraic curve depends on its (topological) genus. Sphere with 3 holes, g = 3 For Fermat curve F p , genus is 1 2 ( p − 1)( p − 2) . ⇒ g = 0 ; • F 2 = ⇒ g = 1 ; Normal forms of Weierstraß and Jacobi + Dixon; • F 3 = • F 4 = ⇒ g = 3 , . . . 7

A clever generalization of sin , cos : the nonlinear system s ′ = c 2 , c ′ = − s 2 with s (0) = 0 , c (0) = 1 . We have: s ( z ) 3 + c ( z ) 3 = 1 : the pair � s ( z ) , c ( z ) � parametrizes F 3 . Follow Dixon (1890) and set: sm( z ) ≡ s ( z ) , cm( z ) ≡ c ( z ) . (See sn, cn by Jacobi, sl, cl for lemniscate.) z − 4 z 4 4! + 160 z 7 7! − 20800 z 10 10! + 6476800 z 13  sm( z ) = 13! − · · ·   1 − 2 z 3 3! + 40 z 6 6! − 3680 z 9 9! + 8880000 z 12 cm( z ) = 12! − · · · .   8

Alfred Cardew Dixon Born: 22 May 1865 in Northallerton, Yorkshire, England Died: 4 May 1936 in Northwood, Middlesex, England Generally, ACD considers X 3 + Y 3 − 3 αXY = 1 . 9

2.1 A hypergeometric connection. One can make s ≡ sm and c ≡ cm somehow “explicit”. Start from the defining system and differentiate √ s ′ = c 2 s ′′ = 2 cc ′ s ′′ = − 2 cs 2 s ′′ = − 2 c ∂ E E = ⇒ = ⇒ = ⇒ s ′ . √ s ′ to integrate ( � Then “cleverly” multiply by ): s ′′ √ 2 3( s ′ ) 3 / 2 = − 2 R 3 s 3 + K. s ′ = − 2 s 2 s ′ = ⇒ � sm( z ) dt � 1 − sm( z ) 3 (1 − t 3 ) 2 / 3 = z, cm( z ) = 3 0 = Abelian integral over F 3 + incomplete Beta integral + hypergeometric 10

Classical hypergeometric function: z 2 2 F 1 [ α, β, γ ; z ] := 1 + α · β 1! + α ( α + 1) · β ( β + 1) z 2! + · · · . γ γ ( γ + 1) Inv( f ) is the inverse of f w.r.t. composition: Inv( f ) = g if f ◦ g = g ◦ f = Id . Proposition: Function sm is defined by inversion, � z � 1 � dt 3 , 2 3 , 4 3; z 3 sm( z ) = Inv (1 − t 3 ) 2 / 3 = Inv z · 2 F 1 . 0 � 1 − sm 3 ( z ) . The function cm is then defined near 0 by cm( z ) = 3 11

3 A STARTLING FRACTION. From Eric van Fossen C ONRAD , PhD Columbus, OH, 2002. Z ∞ x 2 sm( u ) e − u/x du = , 1 · 2 2 · 3 2 · 4 x 6 0 1 + b 0 x 3 − 4 · 5 2 · 6 2 · 7 x 6 1 + b 1 x 3 − 7 · 8 2 · 9 2 · 10 x 6 1 + b 2 x 3 − ... b n = 2(3 n + 1)((3 n + 1) 2 + 1) . where 12

Proof: Follow Stieltjes and Rogers. Cleverly introduce � ∞ sm n ( u ) e − u/x du. S n := 0 Then integration by parts shows that n ( n − 1)( n − 2) x 3 S n = . 1 + 2 n ( n 2 + 1) x 3 − n ( n + 1)( n + 2) x 3 S n +3 S n − 3 S n = ⇒ “Pump” out the continued fraction. Six J -fractions: sm , sm 2 , sm 3 , cm , cm · sm , cm · sm 2 : 1 , 1 , 2 , 2 , 3 , 3 , 4 , 4 , 5 , 5 , 6 , 6 , 7 , 7 , 8 , 8 , . . . + Three S -fractions: sm , cm , sm · cm . 13

4 BALLS GAMES Cf. Th´ eorie analytique des probabilit´ es Laplace (1812). olya urn model. An urn contains black and white balls. At each P´ epoch, a ball in the urn is chosen at random. Described by a placement matrix. Here: 0 1 ❣ − ① ① @ − 1 2 → A , M 12 = ① − ❣ ❣ 2 − 1 → A history of length n [Franc ¸ on78] is any description of a legal sequence of n moves of the P´ olya urn. For instance ( n = 5 ): x − → yy − → yxx − → yyyx − → xxyyx − → xyyyyx , What are the “history numbers”? The sequence for (1 , 0) �→ (0 , ⋆ ) starts as 0 , 1 , 0 , 0 , 4 , 0 , 0 , 160 . Cf sm ? 14

4.1 Urns and Dixonian functions. Take the (autonomous, nonlinear) ordinary differential system dx dy dt = y 2 , dt = x 2 , with Σ : x (0) = x 0 , y (0) = y 0 , � x ( t ) , y ( t ) � parameterizes the “Fermat hyperbola”: y 3 − x 3 = 1 . For x 0 = 0 , y 0 = 1 , get trivial variants: smh( z ) = − sm( − z ) , cmh( z ) = cm( − z ) . 15

Define a linear transformation δ acting on polynomials C [ x, y ] : δ [ x ] = y 2 , δ [ y ] = x 2 , δ [ u · v ] = δ [ u ] · v + u · δ [ v ] , (Cf the elegant presentation of Chen grammars by [Dumont96] and the “com- binatorial integral calculus” of Leroux–Viennot.) ( i ) Combinatorially , the n th iterate δ n [ x a y b ] is such that # histories from ( a 0 , b 0 ) to ( k, ℓ ) = coeff[ x k y ℓ ] δ n [ x a 0 y b 0 ] , ( ii ) Algebraically , the operator δ describes the “logical conse- x = y 2 , ˙ y = x 2 } : quences” of the differential system Σ = { ˙ δ n [ x a y b ] = d n dt n x ( t ) a y ( t ) b expressed in x ( t ) , y ( t ) , ⇒ H ( x ( t ) , y ( t ); z ) = x ( t + z ) a 0 y ( t + z ) b 0 ; set t = 0 . . . ♥ Taylor = 16

♥♥♥ ♥♥♥ Combinatorial Interpretation I Proposition: The EGFs of histories of the urn M 12 starting with one ball: and ending with balls . . . All of the other colour: sm( z ) cm( z ) = − sm( − z ) . 1 All of the original colour: cm( z ) = cm( − z ) . Homogeneous monomial differential systems ⇐ ⇒ k × k balanced urns . Note: Get full composition [=Gaussian], large deviations, etc. √ « 3 3 „ 1 P ( X n = 0) ∼ cρ − n , ρ = 6 π Γ n ≡ 1 (mod 3) . , 3 17

Note: The knight’s moves of Bousquet-Melou & Petkovˇ sek. o----. o----. | | | | . . multiplicity p | | | | P=(p,q) o----.----. P o----.----. | | multiplicity q | | o o The OGF of walks that start at (1 , 0) and end on the horizontal axis is ” 2 ( − 1) i “ X ξ � i � ( x ) ξ � i +1 � ( x ) G ( x ) = , i ≥ 0 x 3 m “ 3 m where ξ , a branch of the (genus 0) cubic xξ − x 3 − ξ 3 = 0 is ξ ( x ) = x 2 X ” 2 m + 1 . m m ≥ 0 18

4.2 Continuous-time branching = Yule process. Foatons and Viennons live an exponential time and disintegrate. . . �→ ; �→ Proposition. Consider the Yule process with two types of parti- cles. The probabilities that particles are all of the second type at time t are X ( t ) = e − t smh(1 − e − t ) , Y ( t ) = e − t cmh(1 − e − t ) , depending on whether the system at time 0 is initialized with one particle of the first type ( X ) or of the second type ( Y ). 19

Remarks on urn processes. For urn � � − α β , − α + β = γ − δ, γ − δ associate a partial differential operator: Γ = x 1 − α y β ∂ ∂x + x γ y 1 − δ ∂ ∂y . ❀ Develop a general theory of P´ olya Urn Processes [FlDuPu06]. ❀ Can characterize all six matrices such that e z Γ is expressible by elliptic functions [FlGaPe05]. One such model ∈ { sm , cm } . 20

„ − 2 3 „ − 1 2 „ − 1 2 « « « A = B = C = , , , 4 − 3 3 − 2 2 − 1 „ − 1 3 „ − 1 3 „ − 1 4 « « « D = , E = , F = . 3 − 1 5 − 3 5 − 2 21