[Paper Review] Taylor Domination, Tur\'an lemma, and Poincar\'e-Perron Sequences

Dmitry Batenkov, Yosef Yomdin|arXiv (Cornell University)|Jan 25, 2013

Stability and Controllability of Differential Equations6 references3 citations

TL;DR

This paper establishes explicit Taylor domination bounds for generating functions of solutions to linear recurrence relations of Poincaré type and for Stieltjes transforms of piecewise D-finite functions. It proves that such functions satisfy uniform Taylor domination in a disk of convergence determined by the spectral properties of the associated recurrence matrix, with explicit control over coefficient decay via subexponential sequences, extending classical results like Turán's inequality and providing a framework for zero counting in analytic functions.

ABSTRACT

We consider "Taylor domination" property for an analytic function $f(z)=\\sum_{k=0}^{\\infty}a_{k}z^{k},$ in the complex disk $D_R$, which is an inequality of the form \\[ |a_{k}|R^{k}\\leq C\\ \\max_{i=0,\\dots,N}\\ |a_{i}|R^{i}, \\ k \\geq N+1. \\] This property is closely related to the classical notion of "valency" of $f$ in $D_R$. For $f$ - rational function we show that Taylor domination is essentially equivalent to a well-known and widely used Tur\\'an's inequality on the sums of powers. Next we consider linear recurrence relations of the Poincar\\'e type \\[ a_{k}=\\sum_{j=1}^{d}[c_{j}+\\psi_{j}(k)]a_{k-j},\\ \\ k=d,d+1,\\dots,\\quad\ ext{with }\\lim_{k\ ightarrow\\infty}\\psi_{j}(k)=0. \\] We show that the generating functions of their solutions possess Taylor domination with explicitly specified parameters. As the main example we consider moment generating functions, i.e. the Stieltjes transforms \\[ S_{g}\\left(z\ ight)=\\int\\frac{g\\left(x\ ight)dx}{1-zx}. \\] We show Taylor domination property for such $S_{g}$ when $g$ is a piecewise D-finite function, satisfying on each continuity segment a linear ODE with polynomial coefficients.

Motivation & Objective

• To establish uniform Taylor domination for solutions of linear recurrence relations of Poincaré type, where coefficients converge to constants.
• To connect Taylor domination with classical inequalities such as Turán's lemma and the Bieberbach conjecture via recurrence structure.
• To analyze the Stieltjes transform of piecewise D-finite functions and prove its Taylor domination with explicit radius and decay control.
• To provide a framework for bounding the number of zeros of generating functions using coefficient domination and spectral properties of recurrence matrices.
• To extend results from rational functions and constant-coefficient recurrences to non-autonomous, asymptotically constant recurrences with explicit error control.

Proposed method

• Use of Taylor domination as a bound on Taylor coefficients via initial coefficients and a subexponential sequence S(k), with explicit control over R and N.
• Equivalence of Taylor domination for rational functions to Turán's inequality on power sums, proven via Biernacki's theorem and Bézout's bound on rational function zeros.
• Analysis of linear recurrence relations of Poincaré type: a_k = ∑_{j=1}^d [c_j + ϕ_j(k)] a_{k-j} with ϕ_j(k) → 0 as k → ∞.
• Construction of a block companion matrix A from the recurrence, whose eigenvalues determine the radius of convergence R* via min{|ξ|^{-1} : ξ ∈ Z_A}.
• Application of spectral theory to the matrix A to derive convergence radius R* and uniform Taylor domination in the disk |z| < R*.
• Use of moment generating functions S_g(z) = ∫ g(x)/(1 - zx) dx for g piecewise D-finite, with recurrence derived from differential operators D of Fuchsian type.

Experimental results

Research questions

• RQ1 Does Taylor domination hold uniformly for solutions of Poincaré-type recurrence relations with asymptotically constant coefficients, and can the radius and decay be explicitly bounded?
• RQ2 Can the classical Turán lemma on power sums be derived as a consequence of Taylor domination for rational functions?
• RQ3 For Stieltjes transforms of piecewise D-finite functions, what is the precise radius of convergence and does Taylor domination hold with explicit parameters?
• RQ4 How does the number of zeros of the generating function relate to the spectral properties of the recurrence matrix A?
• RQ5 Can uniform Taylor domination be established for moment-generating functions m_k = ∫ P^k(x)q(x)dx with polynomial P and q, depending only on degrees?

Key findings

• For solutions of Poincaré-type recurrences with uniformly bounded coefficients, Taylor domination holds with explicit parameters depending only on the size of the perturbations ϕ_j(k).
• The radius of convergence R* for the generating function satisfies R* ≥ min{|ξ|^{-1} : ξ ∈ Z_A}, where Z_A is the set of eigenvalues of the recurrence matrix A.
• For Stieltjes transforms S_g(z) of piecewise D-finite g, Taylor domination holds in the disk |z| < R* with R* determined by the spectral radius of the recurrence matrix A.
• If the differential operator D is Fuchsian and g is not identically zero, then the first Λ(D) + 1 + d_n - n moments cannot vanish, where Λ(D) is the largest positive characteristic exponent at infinity.
• The generating function S_g(z) satisfies (N, R, S(k))-Taylor domination with S(k) subexponential and R = R*, for all N ≥ max{τ - 1, Λ(D)} + d_n - n.
• The paper provides a constructive proof of Turán's inequality via Taylor domination and classical results on multivalent functions and rational function zeros.