Asymptotics of a family of binomial sums

Transcription

1 The binomial sums of inteest Depatment of Mathematics and Statistics Dalhousie Univesity CNTA XI July 16, 2010

2 Outline The binomial sums of inteest 1 The binomial sums of inteest Motivation The esults 2 The multivaiate method of Pemantle and Wilson The tansfe method of Flajolet and Sedgewic

3 The poblem The binomial sums of inteest Motivation The esults Conjectue (Chambeland and Dilche) The sequence a 0 p 1q a dα? 1 c 1 2 satisfies c dα? 1 c 1 c as Ñ 8, fo some d, c 1, c 2, P C, whee α 13 i7? 7 8. Poblem: Study the asymptotics of the binomial sums u pε,a,dq ŗ 0 p 1q ε fo ε P t0, 1u and a, d P N as Ñ 8. a d,

4 Examples The binomial sums of inteest Motivation The esults Cental binomial coefficients: 2 u p0,1,1q Cental Delannoy numbes: Dp, q u p0,1,2q ŗ 0 ŗ 0 2 ; 2 2 ; Binomial sum consideed by Chambeland and Dilche: u p1,2,1q ŗ 0 p 1q 2.

5 Results: Case I The binomial sums of inteest Motivation The esults α 1 p 1q ε d, apα 1qgpzq αz 2 paα a α 1qz 1. g - the disciminant of g z 0 - A paticulaly chosen oot of g. δ 1 p1 z 0 q 4? g, Theoem ( g 0 Case) a β 1 1 αz0 z 0 1 z 0 Thee exist constants µ l such that ŗ 0 p 1q ε a d? 1 δβ 2π µ l l p Ñ 8q

6 Results: Case II The binomial sums of inteest Motivation The esults α 1 p 1q ε d, apα 1qgpzq αz 2 paα a α 1qz 1. g - the disciminant of g z 0 - A paticulaly chosen oot of g. δ 1 p1 z 0 q 4? g, a β 1 1 αz0 z 0 1 z 0 Theoem ( g 0 Case) Thee exist constants µ l such that, as Ñ 8, ŗ a p 1q d ε? δβ µ l 1 2π l 0 δβ? 2π 1 µ l l.

7 The binomial sums of inteest Results: Case III (1 of 2) Motivation The esults Theoem ( g 0 Case) Thee exist constants µ l, η l P Q with denominatos divisible only by the pimes 2 and 3 such that ŗ 2 p 1q 8 p 27q µ l 1 2 2{3 Γp2{3q 1{3 l 0 p 27q 1 2 4{3 Γp1{3q 2{3 η l l p Ñ 8q.

8 The binomial sums of inteest Results: Case III (2 of 2) Motivation The esults Theoem ( g 0 Case) Thee exist constants µ l, η l P Q such that ŗ 0 p 1q {3 p 16q 3Γp2{3q 1{3 2 1{3 p 16q 1 3Γp1{3q 2{3 1 η l l µ l l p Ñ 8q.

9 The binomial sums of inteest Genealized Riodan Aays The multivaiate method of Pemantle and Wilson The tansfe method of Flajolet and Sedgewic Definition ta s u,s is a genealized Riodan aay if F pz, wq,s 0 a s z w s ϕpzq 1 wνpzq fo meomophic functions ϕ and ν that ae analytic at z 0. Lemma Fo ũ s 0 p 1qε as d 0 as p1 αq we can tae ϕpzq 1 a 1 αz 1 z, νpzq. 1 z

10 The binomial sums of inteest The multivaiate method of Pemantle and Wilson The tansfe method of Flajolet and Sedgewic Main esult of Pemantle and Wilson used Fpz, wq,s 0 a sz w s ϕpzq 1 wνpzq, Q spzq z2 ν 2 pzq νpzq p sq. s 2 A pole pz, wq is minimal if fo evey pole pz 1, w 1 q, z 1 z and w 1 w imply z 1 z and w 1 w. S s tz P C pz, νpzq 1 q is minimal, ϕpzq 0, szν 1 pzq νpzq, and szν 2 pzq p sqν 1 pzqu. Poposition (Pemantle and Wilson) Unde suitable conditions and if S s is finite and nonempty then thee exists an asymptotic expansion ϕpz sqνpz a s sqs c pzsq l 1 zsa 2πsQs pz s q s l z sps s as, s Ñ 8 (with {s, s{ emaining bounded), whee? denotes the pincipal banch of the squae oot.

11 The binomial sums of inteest The multivaiate method of Pemantle and Wilson The tansfe method of Flajolet and Sedgewic Classifying the set S s In ou case: S s tz P C pz, νpzq 1 q is minimal and αz 2 pp1 αq p1 αqsqz 0u. a νpzq 1 1 z 1 αz γpzq a fo γpzq 1 z 1 αz. Using the fact that Möbius tansfomations send cicles to cicles we can classify the minimal points. We end up being able to apply the esult of Pemantle and Wilson in all but finitely many cases.

12 The binomial sums of inteest Dealing with the emaining cases The multivaiate method of Pemantle and Wilson The tansfe method of Flajolet and Sedgewic Find the singulaities of the geneating function having least nonzeo modulus. The geneating function satisfies paametic equations and we can diffeentiate implicitly to find the singulaities. Expand the geneating function about these singulaities. A linea homogeneous ODE with polynomial coefficients satisfied by the geneating function can be used fo this. An algebaic equation satisfied by the geneating function can also be used fo this. Tansfe the asymptotic expansion of the geneating function ove to its coefficient sequence by the tansfe method of Flajolet and Sedgewic.

13 The binomial sums of inteest The multivaiate method of Pemantle and Wilson The tansfe method of Flajolet and Sedgewic The main esult of Flajolet and Sedgewic Poposition (Flajolet, Sedgewic) Suppose that ζ 1,..., ζ n ae the dominant singulaities of the odinay geneating function F of the sequence ta u. Unde cetain conditions if F admits an expansion of the fom F pzq c j, pζ j zq θ pz Ñ ζ j q, j fo all j then a ņ j1 c j,j θ 1 ζ j θ j Γpθ j q 1 l j 1 µ j,l l p Ñ 8q whee µ j,l P Qpθ, ζ j, c j,j 1{c j,j,..., c j,l {c j,j q fo each j and l.

14 The binomial sums of inteest The multivaiate method of Pemantle and Wilson The tansfe method of Flajolet and Sedgewic Divisibility popeties of the asymptotic coefficients Stoll and Haible developed a method that can find divisibility popeties of the asymptotic coefficients in cetain asymptotic expansions. Applying the method we obtain the following esult. Poposition Let d P N. Thee exists an asymptotic expansion ŗ 2 d p? d 1q 2 1 µ l 2d 1{4? 1 π l 0 p Ñ 8q, whee the constants µ l P Qp? dq ae such that the only pimes of Qp? dq that can divide thei denominatos ae the pime divisos of 2 and the pime divisos of? d.

15 The binomial sums of inteest The sequences 0 p 1qε a d fo ε P t0, 1u and a, d P N admit full asymptotic expansions as Ñ 8. The main tems can be given explicitly in all cases. On an individual basis, a field containing the asymptotic coefficients can be found, and fo the case ε 0 and a 1, the divisibility popeties of the asymptotic coefficients can be found. Open Questions. Can a numbe field containing the asymptotic coefficients be found in geneal? Can the divisibility popeties of the asymptotic coefficients be found in geneal?