[Paper Review] Newman-Rivlin asymptotics for partial sums of power series

Antonio R. Vargas|arXiv (Cornell University)|Mar 14, 2015

Mathematical Analysis and Transform Methods10 references3 citations

TL;DR

This paper establishes a general asymptotic formula for the ratio of partial sums of entire functions to their limit functions, extending Newman-Rivlin and Edrei-Saff-Varga results. Using a Riemann-Hilbert formulation, it proves that for entire functions of positive finite order with a single direction of maximal growth, the partial sums converge to a complementary error function form, verifying part of the Saff-Varga Width Conjecture in the direction of maximal exponential growth.

ABSTRACT

We discuss analogues of Newman and Rivlin's formula concerning the ratio of a partial sum of a power series to its limit function and present a new general result of this type for entire functions with a certain asymptotic character. The main tool used in the proof is a Riemann-Hilbert formulation for the partial sums introduced by Kriecherbauer et al. This new result makes some progress on verifying a part of the Saff-Varga Width Conjecture concerning the zero-free regions of these partial sums.

Motivation & Objective

• To extend Newman-Rivlin and Edrei-Saff-Varga asymptotic results to a broader class of entire functions with positive finite order.
• To provide a general framework for analyzing the asymptotic behavior of partial sums in the direction of maximal exponential growth.
• To verify a key component of the Saff-Varga Width Conjecture concerning zero-free regions of partial sums.
• To establish conditions under which the partial sum ratio converges uniformly to a complementary error function expression.
• To investigate the limitations of the method in cases with multiple or opposing directions of maximal growth.

Proposed method

• Uses a Riemann-Hilbert formulation for partial sums introduced by Kriecherbauer et al. to analyze asymptotic behavior.
• Applies contour deformation and integral representation techniques to decompose the partial sum into main and error terms.
• Employs a local parametrix construction near the critical point s=1 using a local coordinate system s=ψ(it).
• Implements a splitting of the contour integral into a neighborhood of s=1 and the remainder to control decay rates.
• Uses the assumption that f has a single direction of maximal exponential growth to simplify the phase function φ(s) and control the behavior of the integrand.
• Relies on uniform estimates of the error terms involving δ(rₙ,s) and δ̃(rₙ,s), showing they vanish as n→∞ under the given conditions.

Experimental results

Research questions

• RQ1 Can Newman-Rivlin-type asymptotics be generalized beyond the exponential and Mittag-Leffler functions to a broader class of entire functions?
• RQ2 Under what conditions does the ratio of a partial sum to its limit function converge to a complementary error function form?
• RQ3 How does the asymptotic behavior of partial sums relate to the zero-free regions predicted by the Saff-Varga Width Conjecture?
• RQ4 What are the limitations of the Riemann-Hilbert method when the function has maximal growth in multiple or opposing directions?
• RQ5 Can the method be adapted to functions like f(z) = ∫₋₁¹(1−t)e^{zt}dt, which exhibit maximal growth in opposite directions?

Key findings

• The paper proves that for entire functions of positive finite order λ with a single direction of maximal growth, the partial sum ratio converges uniformly to a complementary error function expression in the complex plane.
• The convergence is established via a Riemann-Hilbert analysis, showing that the main contribution comes from a local parametrix near s=1, with error terms vanishing as n→∞.
• The result verifies part (b) of the modified Saff-Varga Width Conjecture for the case λ=1, m=2, θ=0, with ρₙ=n, confirming the existence of dense zero clusters in the specified region.
• The method fails for functions with maximal growth in two opposite directions, such as f(z) = (e^z - e^{-z}(1+2z))/z², due to geometric obstructions in the Szegő curve's behavior at z=1.
• The asymptotic formula is shown to hold uniformly on compact subsets of Re(w) < 0, with the error terms decaying exponentially under the stated conditions.
• The key technical result is that Fₙ(z) = Gₙ(z) + o(1) as n→∞, where Gₙ(z) captures the leading-order behavior via the complementary error function.