[Paper Review] On zeros of polynomials and allied functions satisfying second order differential equations

Ilia Krasikov|arXiv (Cornell University)|Nov 12, 2002

Mathematical functions and polynomialsMathematics9 references19 citations

TL;DR

This paper establishes sharp, uniform bounds on the extreme zeros of classical orthogonal polynomials—Hermite, Laguerre, and Jacobi—by analyzing functions in the Laguerre-Pólya class that satisfy second-order linear differential equations. Using the logarithmic derivative and discriminant of the differential equation, it derives explicit inequalities for zero spacing, achieving precision comparable to asymptotic expansions, particularly in the large-degree limit.

ABSTRACT

We shall give bounds on the spacing of zeros of certain functions belonging to the Laguerre-Polya class and satisfying a second order differential equation. As a corollary we establish new sharp inequalities on the extreme zeros of the Hermite, Laguerre and Jacobi polinomials, which are uniform in all the parameters.

Motivation & Objective

• To derive new sharp inequalities for the extreme zeros of classical orthogonal polynomials that are uniform across all parameters.
• To extend the method of Laguerre’s inequality on the logarithmic derivative to functions satisfying second-order differential equations with variable coefficients.
• To provide quantitative bounds on zero spacing for polynomials and entire functions in the Laguerre-Pólya class.
• To compare the derived bounds with known asymptotic expansions and existing inequalities, particularly in the large-degree limit.
• To demonstrate the method’s effectiveness on Hermite, Laguerre, Jacobi, and Bessel-type functions, showing convergence to asymptotic values.

Proposed method

• The method relies on the Laguerre-Pólya class condition, which ensures real zeros and a specific canonical product representation.
• It uses the logarithmic derivative $ t(x) = f'(x)/f(x) $ and the discriminant $ \Delta(x) = b(x) - a^2(x) $ of the differential equation $ f'' - 2a f' + b f = 0 $.
• By analyzing intersections of $ a(x) $ with the decreasing branches of $ t(x) $, the paper derives bounds on zero locations via the inequality $ (c - x_j)^2 \Delta(c) \geq 1 $ at intersection points $ c $.
• For extreme zeros, the bounds are optimized over the region $ \Delta(x) > 0 $, leading to expressions like $ x_1 > \min_{x \in J} \left\{ x + \frac{1}{\sqrt{\Delta(x)}} \right\} $.
• The approach is extended to higher-order perturbations using auxiliary functions $ g(x) = f(x)/((x - x_i)(x - x_j)) $, leading to stronger bounds involving $ \Delta(x) $.
• Asymptotic approximations are derived by expanding $ \Delta(x) $ near its zeros, yielding expressions involving $ \Delta'({\bf y}_1)^{-1/3} $, which match known Airy-function asymptotics.

Experimental results

Research questions

• RQ1 Can sharp, uniform bounds on extreme zeros of classical orthogonal polynomials be derived without relying on asymptotic expansions?
• RQ2 How accurately can the spacing of zeros in Laguerre-Pólya class functions be estimated using only the differential equation coefficients?
• RQ3 To what extent do the derived bounds match known asymptotic expansions, especially for large degrees?
• RQ4 Can the method be generalized to entire functions and special functions like Bessel functions?
• RQ5 What is the precision of the bounds relative to the true extreme zeros, particularly in parameter regimes where only the first asymptotic term is known?

Key findings

• The largest zero $ x_M $ of the Hermite polynomial $ H_k(x) $ satisfies $ x_M < \sqrt{2k} - 3 \cdot 2^{-11/6} k^{-1/6} + O(k^{-1/2}) $, closely matching the known asymptotic value $ 1.85575... $.
• For Jacobi polynomials with $ \alpha/k = A $, $ \beta/k = B $ fixed, the extreme zeros satisfy $ x_M < T - 3(1 - T^2)^{2/3}/(2((1+A)(1+B)(1+A+B))^{1/6}) k^{-2/3} + O(k^{-1}) $, with explicit expressions for $ T $.
• The bound for Chebyshev polynomials yields $ x_M < 1 - 0.72 k^{-2} $, while the true value is $ \approx 1 - 1.23 k^{-2} $, showing the method captures the correct order of magnitude.
• For the ultraspherical case $ \alpha = \beta $, the bound gives $ x_M < \frac{\sqrt{1+2A}}{1+A} - \frac{3A^{4/3}}{2(1+A)^{5/3}(1+2A)^{1/6}} k^{-2/3} + O(k^{-1}) $.
• The method’s bounds on zero spacing are shown to improve with higher-order auxiliary functions, with $ c(n) $ for the correction term approaching $ 1.85575... $ as $ n $ increases in $ g = \sum_{i=0}^n (-\lambda)^i \binom{n}{i} f^{(i)} $.
• The derived inequalities are sharp in the sense that they match the leading asymptotic behavior of extreme zeros, particularly in the $ k^{-2/3} $ scaling regime, and are competitive with known asymptotic results.