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Generalized Gauss-Rys orthogonal polynomials

Juan C. García-Ardila, Francisco Marcellán

Abstract

Let $(P_n(x;z;λ))_{n\geq 0}$ be the sequence of monic orthogonal polynomials with respect to the symmetric linear functional $\mathbf{s}$ defined by $$\langle\mathbf{s},p\rangle=\int_{-1}^1 p(x)(1-x^2)^{(λ-1/2)} e^{-zx^2}dx,\qquadλ>-1/2, \quad z>0.$$ In this contribution, several properties of the polynomials $P_n(x;z;λ)$ are studied taking into account the relation between the parameters of the three-term recurrence relation that they satisfy. Asymptotic expansions of these coefficients are given. Discrete Painlevé and Painlevé equations associated with such coefficients appear naturally. An electrostatic interpretation of the zeros of such polynomials as well as the dynamics of the zeros in terms of the parameters $z$ and $λ$ are given.

Generalized Gauss-Rys orthogonal polynomials

Abstract

Let be the sequence of monic orthogonal polynomials with respect to the symmetric linear functional defined by In this contribution, several properties of the polynomials are studied taking into account the relation between the parameters of the three-term recurrence relation that they satisfy. Asymptotic expansions of these coefficients are given. Discrete Painlevé and Painlevé equations associated with such coefficients appear naturally. An electrostatic interpretation of the zeros of such polynomials as well as the dynamics of the zeros in terms of the parameters and are given.
Paper Structure (9 sections, 18 theorems, 136 equations)

This paper contains 9 sections, 18 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Basic background
  3. Generalized Gauss-Rys orthogonal polynomials
  4. Laguerre-Freud's equations
  5. Holonomic Differential equation
  6. Variable $z$
  7. Zeros
  8. Electrostatic interpretation
  9. Dynamical behavior of zeros

Key Result

Proposition 2.5

Let $\mathbf{u}$ be a semi-classical 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, $\theta_c f(x)=\dfrac{f(x)-f(c)}{x-c}.$

Theorems & Definitions (42)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: Ma87
  • Definition 2.4
  • Proposition 2.5: GMM21Ma91
  • Theorem 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • ...and 32 more