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<p>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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13660_2025_3283_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$(-1,1)$</EquationSource> </InlineEquation>. 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.</p>

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Uniform relations between the Gauss–Legendre nodes and weights

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年3月20日

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  • Óscar López Pouso,
  • Javier Segura

摘要

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 ) $(-1,1)$ . 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.