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An electrostatic model for the roots of discrete classical orthogonal polynomials

Joaquín F. Sánchez-Lara

Abstract

An electrostatic model is presented to describe the behaviour of the roots of classical discrete orthogonal polynomials. Indeed, this model applies in the more general frame of polynomial solutions of second-order linear difference equations $$AΔ_h\nabla_h y+BΔ_h y+ C y=0\,,$$ where $A$, $B$ and $C$ are polynomials and $$Δ_h f(x)=f(x+h)-f(x)\qquad \text{ and }\qquad \nabla_h f(x)=f(x)-f(x-h)$$ with $h>0$.

An electrostatic model for the roots of discrete classical orthogonal polynomials

Abstract

An electrostatic model is presented to describe the behaviour of the roots of classical discrete orthogonal polynomials. Indeed, this model applies in the more general frame of polynomial solutions of second-order linear difference equations where , and are polynomials and with .

Paper Structure

This paper contains 9 sections, 8 theorems, 104 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Electrostatic model
  3. Application to orthogonal polynomials in the Askey scheme
  4. Charlier polynomials
  5. Krawtchouk polynomials
  6. Meixner polynomials
  7. Hahn polynomials
  8. Dual Hahn polynomials
  9. Racah polynomials

Key Result

Theorem 2.4

Consider two polynomials $A$ and $B$ and an interval $(a,b)$ in which $1+B/A>0$.

Figures (2)

  • Figure 1: Left: Force field given by $F_0$ with $h=1$ defined by \ref{['Forceh']} in blue and force field $F_0^{\log}$ defined by \ref{['ForceLog']} in black. Right: Rate between the two force fields.
  • Figure 2: Potentials $V_0^{\text{log}}$ in black and $V_0$ with $h=1$ in blue that an unitary charge at $x=0$ creates.

Theorems & Definitions (20)

  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • proof
  • Definition 2.5
  • Remark 2.6
  • Theorem 2.8
  • proof
  • Corollary 2.9
  • ...and 10 more