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A class of Truncated Freud polynomials

Juan Carlos García-Ardila, Francisco Marcellán, Misael E. Marriaga

TL;DR

The paper addresses the orthogonal polynomial system for the truncated Freud weight u_z(p) with p on [0, ∞) weighted by e^{-z x^4}, z>0, characterizing its semiclassical nature and deriving a complete set of structural tools. The authors establish that u_z is semiclassical of class s=3, derive Laguerre-Freud nonlinear difference equations for the recurrence coefficients, obtain a holonomic second-order differential equation for the polynomials, and provide an electrostatic interpretation of their zeros, together with detailed numerical experiments. Key contributions include explicit moment formulas μ_n(z) = (z^{−(n+1)/4}/4) Γ((n+1)/4), a closed nonlinear system for the TTRR coefficients a_n and b_n with precise asymptotics a_n ~ √(n/(140 z)) and b_n ~ 2 (n/(140 z))^{1/4}, and a fully worked holonomic framework plus a scaled zero distribution in the large-n limit. The work advances understanding of semiclassical truncated Freud polynomials, connects to integrable systems and external-potential interpretations, and provides practical guidance for quadrature and spectral problems on [0, ∞) in Druyvesteyn-type contexts.

Abstract

Consider the following truncated Freud linear functional $\mathbf{u}_z$ depending on a parameter $z$, $$\langle\mathbf{u}_z,p\rangle=\int_0^\infty p(x)e^{-zx^4}dx,\quad z>0.$$ The aim of this work is to analyze the properties of the sequence of orthogonal polynomials $(P_n)_{n\geq 0}$ with respect to $\mathbf{u}_z$. Such a linear functional is semiclassical and, as a consequence, we get the system of nonlinear difference equations (Laguerre-Freud equations) that the coefficients of the three-term recurrence satisfy. The asymptotic behavior of such coefficients is given. On the other hand, the raising and lowering operators associated with such a linear functional are obtained, and thus a second-order linear differential equation of holonomic type that $(P_n)_{n\geq 0}$ satisfies is deduced. From this fact, an electrostatic interpretation of their zeros is given. Finally, some illustrative numerical tests concerning the behavior of the least and greatest zeros of these polynomials are presented.

A class of Truncated Freud polynomials

TL;DR

The paper addresses the orthogonal polynomial system for the truncated Freud weight u_z(p) with p on [0, ∞) weighted by e^{-z x^4}, z>0, characterizing its semiclassical nature and deriving a complete set of structural tools. The authors establish that u_z is semiclassical of class s=3, derive Laguerre-Freud nonlinear difference equations for the recurrence coefficients, obtain a holonomic second-order differential equation for the polynomials, and provide an electrostatic interpretation of their zeros, together with detailed numerical experiments. Key contributions include explicit moment formulas μ_n(z) = (z^{−(n+1)/4}/4) Γ((n+1)/4), a closed nonlinear system for the TTRR coefficients a_n and b_n with precise asymptotics a_n ~ √(n/(140 z)) and b_n ~ 2 (n/(140 z))^{1/4}, and a fully worked holonomic framework plus a scaled zero distribution in the large-n limit. The work advances understanding of semiclassical truncated Freud polynomials, connects to integrable systems and external-potential interpretations, and provides practical guidance for quadrature and spectral problems on [0, ∞) in Druyvesteyn-type contexts.

Abstract

Consider the following truncated Freud linear functional depending on a parameter , The aim of this work is to analyze the properties of the sequence of orthogonal polynomials with respect to . Such a linear functional is semiclassical and, as a consequence, we get the system of nonlinear difference equations (Laguerre-Freud equations) that the coefficients of the three-term recurrence satisfy. The asymptotic behavior of such coefficients is given. On the other hand, the raising and lowering operators associated with such a linear functional are obtained, and thus a second-order linear differential equation of holonomic type that satisfies is deduced. From this fact, an electrostatic interpretation of their zeros is given. Finally, some illustrative numerical tests concerning the behavior of the least and greatest zeros of these polynomials are presented.

Paper Structure

This paper contains 11 sections, 17 theorems, 129 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Basic background
  3. Truncated Freud polynomials
  4. Laguerre-Freud equations
  5. Holonomic differential equation
  6. The role of the variable $z$
  7. Zeros and their electrostatic interpretation
  8. Asymptotic zero distribution
  9. Electrostatic interpretation
  10. Numerical experiments
  11. Concluding remarks.

Key Result

Proposition 2.4

Let $\mathbf{u}$ be a semiclassical linear functional, and let $\phi(x)$ and $\psi(x)$ be non-zero polynomials with $\deg\phi(x)=:r$ and $\deg \psi(x)=:t$, such that pearson-semic is satisfied. Let $s := \max(r-2,t-1)$. Then $s = \mathfrak{s}(\mathbf{u})$ if and only if Here, $(\theta_c f ) (x)=\dfrac{f(x)-f(c)}{x-c}, x\neq c, (\theta_c f ) (c)= f ' (c).$

Figures (5)

  • Figure 1: The limit distribution of zeros $\omega(x,t)$ for different values of $t$
  • Figure 2: Behavior of smallest and largest zero for each polynomial $P_n(x)$, with $z=1$ and $n=1,2\ldots, 14.$
  • Figure 3: Linear trend of the modified smallest and largest zero $x_{n,1}^{-1/2}$ and $x^3_{n,n}$, respectively for $n=1,2,\ldots, 14.$
  • Figure 4: Comparative of $y_{n,1}=\frac{2}{\sqrt[4]{140}}\left(\cos\left(\frac{n\pi}{n+1}\right)+1\right)$ with the smallest zero $x_{n,1}$ for $n=1,\ldots, 14.$
  • Figure 5: Comparative of the largest zeros $x_{n,n}$ and $\widetilde{x}_{n,n}$ for $n=1,\ldots, 14.$

Theorems & Definitions (33)

  • Definition 2.1
  • Definition 2.2: Ma87
  • Definition 2.3
  • Proposition 2.4: GMM21Ma91
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • Corollary 3.2
  • Proposition 3.3
  • proof
  • ...and 23 more