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Uniform relations between the Gauss-Legendre nodes and weights

Óscar López Pouso, Javier Segura

Abstract

Four different relations between the Legendre nodes and weights are presented which, unlike the circle and trapezoid theorems for Gauss-Legendre quadrature, hold uniformly in the whole interval $(-1,1)$. These properties are supported by strong asymptotic evidence. The study of these results was originally motivated by the role some of them play in certain finite difference schemes used in the discretization of the angular Fokker-Planck diffusion operator.

Uniform relations between the Gauss-Legendre nodes and weights

Abstract

Four different relations between the Legendre nodes and weights are presented which, unlike the circle and trapezoid theorems for Gauss-Legendre quadrature, hold uniformly in the whole interval . These properties are supported by strong asymptotic evidence. The study of these results was originally motivated by the role some of them play in certain finite difference schemes used in the discretization of the angular Fokker-Planck diffusion operator.
Paper Structure (17 sections, 15 theorems, 128 equations)

This paper contains 17 sections, 15 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Some preliminary asymptotic results
  3. Elementary expansions
  4. Bessel-type expansions
  5. The circle and trapezoid theorems
  6. A first uniform relation: the uniform circle theorem for Gauss-Legendre quadrature
  7. Uniform relations involving sums of weights
  8. Relation between the primary and the secondary nodes
  9. Relation between the primary and secondary intermediate nodes
  10. Relation between the first order partial moments and the intermediate nodes
  11. A complementary observation that supports Conjecture \ref{['lili4']}
  12. Applications in finite difference schemes
  13. Appendix: Elementary expansion for the weights
  14. Appendix: Bessel-type expansion for the weights
  15. Appendix: proof of a property of Bessel functions
  16. ...and 2 more sections

Key Result

Theorem 1

Let $[a,b]\subset (-1,1)$, then

Theorems & Definitions (36)

  • Theorem 1
  • proof
  • Conjecture 1: Uniform circle theorem
  • Corollary 1
  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 2
  • ...and 26 more