Taylor Domination, Turán lemma, and Poincaré-Perron Sequences Dmitry Batenkov and Yosef Yomdin 3 1 0 2 n a J 5 2 Abstract Weconsiderlinearrecurrence relations ofthePoincarétype ] d A ak =X[cj+ψj(k)]ak−j, k=d,d+1,..., with kl→im∞ψj(k)=0. C j=1 h. Weshowthat theirsolutionsa0,a1,...,ad−1,ad,ad+1,... satisfytheTurán-likeinequal- ity:forcertain N andRandforeachk≥N+1 t a m |ak|Rk ≤C max |ai|Ri. i=0,...,N [ For the generating function f(z) = P∞k=0akzk we interpret this last inequality as the 1 “Taylordomination” property:alltheTaylorcoefficientsoff areboundedthroughthefirst v N ofthem. Wegiveconsequences ofthis property, showingthat f belongs tothe classof 3 (s,p)-valentfunctions,whichformasubclassoftheclassicallystudiedp-valentones. 3 Wealsoconsidermomentgenerating functions,i.e.theStieltjestransforms 0 .6 Sg(z)=ˆ g1(−x)zdxx. 1 0 WeshowTaylordominationproperty forsuchSg wheng isapiecewiseD-finitefunction, 3 satisfyingoneachcontinuitysegmentalinearODEwithpolynomialcoefficients.Weutilize 1 thefactthatthemomentsequence ofg satisfiesa Poincaré-type recurrence. : v i DmitryBatenkov X Department of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel. This r author issupported bythe AdamsFellowshipProgram ofthe Israel Academy of Sciences a andHumanities.e-mail:[email protected] YosefYomdin Department of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel. This author is supported by ISF, Grant No. 639/09 and by the Minerva foundation. e-mail: [email protected] 1 2 DmitryBatenkovandYosefYomdin 1 Introduction The problem of analytic continuation, i.e. of a global reconstruction of a function f and of its analytic properties, from the sequence of its Taylor coefficientsatapoint,isoneofthemostclassicalproblemsinAnalysis.Ittook acentralroleinthe classicalinvestigationsfromthe endofthe nineteenthto the middle of the twentieth centuries. See [4, 5] and references therein for a sample of the classicalresults available. Some of these results are (arguably) considered as the deepest and most difficult results in Classical Analysis. We are interested in a somewhat restricted setting of this problem: namely, the casewhen the sequence ofTaylorcoefficients depends ona finite number of parameters. Definition 1.1.The sequence {a }∞ is said to satisfy a linear recurrence k k=0 relation of Poincaré type (see [16, 18]) if d a = [c +ψ (k)]a , k =d,d+1,..., (1) k j j k−j j=1 X for some fixed coefficients {c ,ψ (k)}d , satisfying j j j=1 lim ψ (k)=0. j k→∞ We also say that the coefficients {c ,ψ (k)}d define an element S of j j j=1 the Poincaré class R. If in addition these coefficients depend on a finite- dimensional parameter vector λ∈Cm, then we write S for the correspond- λ ing recurrence. Given a certain S ∈ R, the solution of (1) is completely determined by λ λ and the initial data a¯ = (a ,a ,...,a ). In this case, we denote by 0 1 d−1 f(z)=f (z)thegeneratingfunctionofthecorrespondingsolution{a }∞ , λ,a¯ k k=0 i.e. the function represented in its disk of convergence by the power series ∞ f (z)= a zk. (2) λ,a¯ k k=0 X The main problem we discuss in this paper is the following. Problem 1.1.To what extent is it possible to read out the global analytic properties of the function f (z) from the parameters λ,a¯? This includes, λ,a¯ in particular, the position and type of singularities of f(z), its ramification properties, and the bounds on the number of zeros of f(z). Our setting, described above, is motivated by two main long-standing open problems in qualitative theory of differential equations: Hilbert’s 16th prob- lem (2nd part) and Poincaré’s Center-Focus problem. A promising approach TaylorDomination,Turánlemma,andPoincaré-Perron Sequences 3 to these problems is based on the fact that the Taylor coefficients of the correspondingPoincaréfirst-returnmapsatisfyacertaindifferential-typere- currence relation. The main question is essentially to obtain information on the zeroes of f. For more details, see e.g. [8]. Ourapproachstressesthe recurrencerelation(1)asthe sourceofthe infinite sequence of the Taylor coefficients. So, in contrast to the classical setting, we would like to read the finite-dimensional global information on f from a finite-dimensional input:the parametersλ,a¯.While alsointhis settingmost of the problems remain very difficult, we can hope to translate at least some of the classical results into the new setting. In this paper we present a small piece of the above program, translating theclassicalresultsonunivalentandmulti-valentfunctionsin[5,13]intothe languageoftherecurrencerelation(1).Theseresultswereobtainedmainlyin connection to the classicalBieberbach conjecture, finally settled in [10]. The property of the Taylor coefficients, treated in [5, 13] is what we call “Taylor domination”: bounding all the Taylor coefficients through the first few ones. See Section 2. The classical Turán’s lemma ([15, 22, 23]) is central in many problems in HarmonicAnalysis.Itprovidesuniform Taylordominationforrationalfunc- tions (i.e. for solutions of the recurrence relation (1) with ψ (k) ≡ 0). In j Section 3 we present a partial extension of the Turán lemma to general re- currence relations (1). As a consequence, we obtain a bound on the number of zeroes of the generating functions (2). In Section 4 we provide a natural settingofthisresultintermsof“(s,p)-valentfunctions”.Thisnotionprovides ageneralizationofthe classicallyconsideredmulti-valentfunctions in[5,13]. Not surprisingly,the problem of a full extension of the Turán lemma to gen- eral Poincaré type recurrence relations (1) is closely connected to the theo- remsofPoincaréandPerronandtheirextensions(see[6,7,14,16,17,18,19] andreferencestherein),whichprovideanaccurateasymptoticbehaviorofthe solutionsof (1).UsingPerron’sSecondTheoremanditsrecentgeneralization byM.Pituk,weshowthatthesolutionsof (1),(evenwithouttheassumption ondependenceonfinitenumberofparameters),satisfytheTaylordomination property.UnderadditionalassumptionsweshowuniformTaylordomination, which results in a bound on number of zeros. See Section 5 for details. Sometimes(forinstanceinthefirst-orderapproximationofthePoincarémap- ping to the Abel equation) the Taylor coefficients of the function f of inter- est are given as linear functionals, such as moments, of some other “simple” functiong.InSection6weconsidermoment-generatingfunctions,orStieltjes transforms f =S (z)= g(x)dx = ∞ m zk, where m = xkg(x)dx for g 1−xz k=0 k k g - piecewiseD-finite fun´ctions,satisfyingoneachtheir contin´uitysegmenta P linear homogeneous ODE with polynomial coefficients. In this case, the mo- mentsequenceofg satisfiesaPoincaré-typerecurrenceS withψ (k)-rational j functions (in k), and we show Taylor domination property for S . g 4 DmitryBatenkovandYosefYomdin 2 Taylor domination 2.1 Definitions and basic facts Definition 2.1.A non-negative sequence S(k) : N → R is said to be of >0 sub-exponential growth if for any ε>0 we have S(k) lim =0. (3) k→∞exp(εk) Let f(z)= ∞ a zk, a ∈C, be a power series with the radius of conver- k=0 k k gence 0<Rˆ 6+∞ . P Definition 2.2.Let a positive finite R ≤ Rˆ, a natural N, and a positive sequence S(k) of sub-exponentialgrowth,be fixed. The function f is saidto possess an (N,R,S(k)) - Taylor domination property if for each k ≥ N +1 we have |a |Rk 6S(k) max |a |Ri. k i i=0,...,N For S(k) ≡ C a constant we shall call this property (N,R,C)-Taylor domi- nation. Proposition 2.1.(N,R,S(k))-dominationimplies(N,R′,C)-dominationfor every 0<R′ <R and C a certain constant depending on R′ and on the se- R quence S(k). Proof. Let ρ d=ef R′ < 1. First, let us show that the sequence ρkS(k) is R bounded: S(k) lim ρkS(k) = lim S(k)exp(klnρ)= lim , k→∞ k→∞ k→∞ exp klnρ−1   >0   and this is zero by (3). | {z } So let def M = supρkS(k). k>0 Finally we have for k >N |a |(R′)k = |a |Rkρk 6ρkS(k) max |a |(R′)iρ−i 6Mρ−N max |a |(R′)i. k k i i i=0,...,N i=0,...,N This finishes the proof with C =Mρ−N. ⊓⊔ TaylorDomination,Turánlemma,andPoincaré-Perron Sequences 5 Lemma 2.1.Let {a }∞ be a sequence satisfying k k=0 limsup k |a |=σ <+∞, (4) k k→∞ p and denote1 by Rˆ d=efσ−1 the radius of convergence of the power series ∞ f = a zk. k k=0 X If f 6≡0, then for each finite and positive 0<R6Rˆ, each N ∈N satisfying a 6=0 and for S(k) defined as: N S(k)d=ef|a |R˜k× max |a |R˜i −1, where R˜ d=ef Rˆ σ >0, k i (cid:26)i=0,...,N (cid:27) (R σ =0 f satisfies (N,R,S(k))-Taylor domination property. Proof. Consider two cases. 1. σ > 0. By Proposition 2.1, it is sufficient to show N,Rˆ,S(k) -Taylor domination. We have for each k >N (cid:16) (cid:17) |a |Rˆk =S(k) max |a |Rˆi, k i i=0,...,N andtherefore the onlything leftto showis thatS(k)is ofsubexponential growth. Denote C d=ef max |a |Rˆi >0. i i=0,...,N Then we have for each ε>0 S(k) |ak|Rˆk 1 |ak|k1 k lim = lim = lim . k→∞exp(εk) k→∞Cexp(εk) C k→∞ σexp(ε)! 1 Denote η = exp(ε) > 1. By (4), starting from some k we have |ak|k < 0 σ 1+ η−1 = 1+η, and therefore 2 2 1 |ak|k 1+η 1 1 < = 1+ <1. ση 2η 2 η (cid:18) (cid:19) As a result, the whole limit exists and is equal to zero. 2. If σ =0, then for each R<∞ we again have 1 Justbyapplyingthestandardroottest. 6 DmitryBatenkovandYosefYomdin |a |Rk =S(k) max |a |Ri, k i i=0,...,N =C and also for every ε>0 | {z } 1 k S(k) 1 |ak|k R lim = lim . k→∞exp(εk) C k→∞ exp(ε)! 1 From the assumption, we actually have limk→∞|ak|k = 0, and therefore the whole limit above exists and equals to zero. This finishes the proof. ⊓⊔ As animmediate corollarywe conclude that the Taylordominationproperty issatisfiedbyanyconvergingpowerseriesinitsmaximaldiskofconvergence. Corollary 2.1.If 0 < Rˆ 6 +∞ is the radius of convergence of f(z) = ∞ a zk, with f 66≡ 0, then for each finite and positive 0 < R 6 Rˆ, f k=0 k satisfies the (N,R,S(k))-Taylor domination property with N and S(k) as P defined in Lemma 2.1. As we showin the next subsection,the notionofTaylordominationis useful for investigating global analytic properties of the corresponding generating functions (in particular, their number of zeros), but only if the constants (N,R,C) are in a sense “as tight as possible”. Therefore the main question related to Taylor domination is finding these optimal constants for certain families of generating functions {f }. λ,a¯ 2.2 Global properties of generating functions and families Applying the roottest, we immediately conclude that the converseto Corol- lary 2.1 is also true. Proposition 2.2.Let f possess (N,R,S(k))-domination. Then the series ∞ a zk k k=0 X converges in a disk of radius R∗ satisfying R∗ >R. TaylorDomination,Turánlemma,andPoincaré-Perron Sequences 7 Taylor domination allows us to compare the behavior of f(z) with the be- havior of the polynomial P (z) = N a zk. In particular, the number of N k=0 k zeroes of f can be easily bounded in this way. P Theorem 2.1([21,Lemma2.2]).ThereexistsafinitefunctionM(N,R′,C), R satisfying lim M =+∞, R′→1 R and equal to N for R′ sufficiently small, such that the following bound holds: R if f possesses a (N,R,C) - Taylor domination property then for any R′ <R, f has at most M(N,RR′,C) zeros in DR′. In particular, put R = R, R = R and R = R . Then 1 4 2 2max(C,2) 3 23Nmax(C,2) the number of zeroes of f in the disks D , D and D does not exceed R1 R2 R3 5N +log (2+C), 5N +10 and N, respectively. 5/4 AnexplicitexpressionforM isgivenin[21,Proposition2.2.2].ByProposition 2.1, the corresponding bounds are true also in a general case of (N,R,S(k)) - Taylor domination. ItisimportanttostressthatTaylordominationpropertyisessentiallyequiv- alent to the bound on the number of zeroes of f −c, for each c. Let us give the following definition (see [13] and references therein). Definition 2.3.A function f regular in a domain Ω ⊂ C is called p-valent there, if for any c∈C the number of solutions in Ω of the equation f(z)=c does not exceed p. Theorem 2.2([5]). If f is p-valent in the disk D of radius R centered at R 0∈C then |a |Rk ≤(Ak/p)2p max |a |Ri. k i i=0,...,p In our notations, Theorem 2.2 claims that a function f which is p-valent in D , possesses an (N,R,(Ak/p)2p) - Taylor domination property. Theorem R 2.1 shows that the inverse is also essentially true. Forunivalentfunctions,i.e.for p=1,andfor a =0, R=1, the Bieberbach 0 conjecture proved by De Branges in [10] claims that |a |≤k|a | for each k. k 1 Let us now consider the families of generating functions (2), taking into ac- count the dependence on the parameter vector λ. Our ultimate goal is to be abletoboundthe numberofzerosof f (z)uniformly in λ.Forthispurpose, λ we introduce the notion of uniform Taylor domination, as follows. Definition 2.4.Let N ∈ N, R(λ) : Cm → R+ and a sequence S(k) : N → R+ of subexponential growth be fixed. The family {f (z)} of generating λ functions (2) is said to possess uniform (N,R(λ),S(k))-Taylor domination property if for each λ, the function f (z) possesses the (N,R(λ),S(k))- λ domination (c.f. Definition 2.2), with N and S(k) independent of λ. 8 DmitryBatenkovandYosefYomdin Consequently, if {f (z)} possesses uniform (N,R(λ),S(k))-Taylor domina- λ tion,then for eachλ the radiusof convergenceoff (z)is at leastR(λ),and λ thenumberofzerosinconcentricdisksDR′ withafixedproportion RR′((λλ)) can be uniformly (in λ) bounded, according to Theorem 2.1. 2.3 Rational generating functions and Turán’s lemma Let us start with the following well-known fact (see e.g. [11, Section 2.3]). Proposition 2.3.Consider a linear recurrence relation S with constant co- efficients d a = c a , k =d,d+1,.... (5) k j k−j j=1 X 1. For any initial data a¯=(a ,a ,...,a ), the general solution of (5) has 0 1 d−1 the form s a = σkP (k), (6) k j j j=1 X where {σ ,...,σ } are the distinct roots of the characteristic equation 1 s d Θ (σ)=σd− c σd−j =0, S j j=1 X with corresponding multiplicities {m ,...,m }, and each P (k) is a poly- 1 s j nomial of degree at most m −1. j 2. The generating function f (z) of S is a rational function of the form S,a¯ P(z) f(z)= , degP (z)6d−1, Q (z) S where d Q (z)=zdΘ z−1 =1− c zj S S j j=1 (cid:0) (cid:1) X and degP (z)6d−1. 3. Conversely,foreach(regularattheorigin)rationalfunctionf(z)= P(z) = Q(z) ∞ a zk,withQ(z)= s (1−zσ )mj and s m =d,degP(z)≤ k=0 k j=1 j j=1 j d−1, its Taylor coefficients a satisfy (5), with {c }d defined by P Q k Pj j=1 TaylorDomination,Turánlemma,andPoincaré-Perron Sequences 9 d Q(z)=1− c zj. j j=1 X ThemostbasicexampleofTaylordomination,concerningrationalfunctions, is provided by the Turán lemma ([22, 23], see also [15]) as follows. Theorem 2.3([22]). Let {a }∞ satisfy the recurrence relation (5). Using j j=1 notations of Proposition 2.3, let {x ,...,x } be the characteristic roots of S, 1 d and put Rd=ef min x−1 . i i=1,...d (cid:12) (cid:12) Then for each k ≥d+1 (cid:12) (cid:12) |a |Rk ≤C(d) kd max |a |Ri. (7) k i i=0,...,d This theorem provides a uniform Taylor domination (Definition 2.4) for ra- tional functions in their maximal disk of convergence D , in the strongest R possible sense. Indeed, after rescaling to D the parameters of (7) depend 1 only on the degree of the function, but not on its specific coefficients. Turán’slemmacanbeconsideredasaresultonexponentialpolynomials2,and inthis formitwasastartingpointformanydeepinvestigationsinHarmonic Analysis,Uncertainty Principle, Analytic continuation,Number Theory (see [15, 22, 23] and references therein). The main problem we investigate in this paper is a possibility to extend a uniform Taylor domination in the maximal disk of convergence D , as pro- R vided by Theorem 2.3 for rational functions, to a wider class of generating functions of Poincaré type recurrencerelations (1). As we show in Section 5, the (N,R,S(k))-domination is always satisfied in the entire disk of conver- gence for solutions of (1), howeverin generalthe domination is not uniform, as the sequence S(k) depends on all the parameters {ψ }∞ , a¯ (in contrast j j=0 to (7)). We shall prove below in Section 3, for linear recurrence relations even more generalthan(1),aweakerversionofTurán’slemma.ItprovidesuniformTay- lor domination not in the maximal disk of convergence,but in its concentric sub-disks of a sufficiently small radius. 2 Indeed, the formula (6) shows that ak are the values of the exponential polynomial φ(t)=Psj=1λtjPj(t)atthepointst∈N. 10 DmitryBatenkovandYosefYomdin 3 Uniform Taylor domination in a smaller disk Firstlet us define the class ofrecurrencerelationsslightlymore generalthan the Poincaré class R. Definition 3.1.A linear recurrence relation S is said to belong to the class Rˆ(K,ρ) (where ρ>0 and K >0 are two fixed parameters), if d a = c (k)a , k =d,d+1,..., k j k−j j=1 X where the coefficients c (k) satisfy for each k ∈N j |c (k)|6Kρj, j =1,...,d. j So we do not require,as in(1), the convergenceofthe recurrencecoefficients for k tending to infinity. In this section we prove Taylor domination for solutions of the recurrence relations in Rˆ(K,ρ). As we show later in Section 5, for each S ∈ R there exist some K,ρ > 0 such that S ∈ Rˆ(K,ρ), and as a result we will obtain Taylor domination for such S with appropriately chosen constants. Theorem 3.1.Let {a }∞ be a solution of the recurrence relation S ∈ k k=0 Rˆ(K,ρ). Put Rd=efρ−1. Then for each k >d we have |a |Rk 6(2K+2)k max |a |Ri. (8) k i i=0,...,d−1 Proof. The proof is by induction on k. Denote def M = max |a |Ri, i i=0,...,d−1 def η = 2K+2, and assume that |a |Rℓ 6ηℓM, ℓ6k−1. (9) ℓ By definition, (9) holds for 06ℓ6d−1. We have d d d |a |Rk =Rk| c (k)a |≤KRk |a |ρj =K |a |Rk−j. k j k−j k−j k−j j=1 j=1 j=1 X X X By the inductive assumption |a |Rk−j 6 ηk−jM, therefore we conclude k−j that