@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr:-DCG8G0KK-F
  skos:exactMatch <https://fr.wikipedia.org/wiki/Polyn%C3%B4me_de_Laguerre>, <https://en.wikipedia.org/wiki/Laguerre_polynomials> ;
  skos:definition """In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are nontrivial solutions of Laguerre's differential equation :
         <br/><div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle xy''+(1-x)y'+ny=0,\\\\ y=y(x)}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>x</mi>
         <msup>
         <mi>y</mi>
         <mo>″</mo>
         </msup>
         <mo>+</mo>
         <mo stretchy="false">(</mo>
         <mn>1</mn>
         <mo>−<!-- − --></mo>
         <mi>x</mi>
         <mo stretchy="false">)</mo>
         <msup>
         <mi>y</mi>
         <mo>′</mo>
         </msup>
         <mo>+</mo>
         <mi>n</mi>
         <mi>y</mi>
         <mo>=</mo>
         <mn>0</mn>
         <mo>,</mo>
         <mtext> </mtext>
         <mi>y</mi>
         <mo>=</mo>
         <mi>y</mi>
         <mo stretchy="false">(</mo>
         <mi>x</mi>
         <mo stretchy="false">)</mo>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle xy''+(1-x)y'+ny=0,\\\\ y=y(x)}</annotation>
         </semantics>
         </math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46eb8b594da7061a7ea73c7ad9f3a0574d3ccb58" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.269ex; height:3.009ex;" alt="{\\\\displaystyle xy''+(1-x)y'+ny=0,\\\\ y=y(x)}"></div>
<br/>which is a second-order linear differential equation. This equation has nonsingular solutions only if n is a non-negative integer. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Laguerre_polynomials">https://en.wikipedia.org/wiki/Laguerre_polynomials</a>)"""@en, """En mathématiques, les <b>polynômes de Laguerre</b>, nommés d'après Edmond Laguerre, 
<br/>sont les solutions normalisées de <b>l'équation de Laguerre</b>&nbsp;:
<br/>
<br/><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle x\\\\,y''+(1-x)\\\\,y'+n\\\\,y=0\\\\,}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>x</mi>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>        <msup>
<br/>          <mi>y</mi>
<br/>          <mo>″</mo>
<br/>        </msup>
<br/>        <mo>+</mo>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mn>1</mn>
<br/>        <mo>−<!-- − --></mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>        <msup>
<br/>          <mi>y</mi>
<br/>          <mo>′</mo>
<br/>        </msup>
<br/>        <mo>+</mo>
<br/>        <mi>n</mi>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>        <mi>y</mi>
<br/>        <mo>=</mo>
<br/>        <mn>0</mn>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle x\\\\,y''+(1-x)\\\\,y'+n\\\\,y=0\\\\,}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adbe33c805a58129e3f1fde424c2e5edd208f743" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.655ex; height:3.009ex;" alt="{\\\\displaystyle x\\\\,y''+(1-x)\\\\,y'+n\\\\,y=0\\\\,}"></span></dd></dl>
<br/>qui est une équation différentielle linéaire homogène d'ordre 2 et se réécrit sous la forme de Sturm-Liouville&nbsp;:
<br/><span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle -{{\\
m {d}} \\\\over {\\
m {d}}x}\\\\left(x{\\
m {e}}^{-x}{{\\
m {d}}y \\\\over {\\
m {d}}x}\\
ight)=n{\\
m {e}}^{-x}y.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mo>−<!-- − --></mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mi mathvariant="normal">d</mi>
<br/>              </mrow>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mi mathvariant="normal">d</mi>
<br/>                </mrow>
<br/>              </mrow>
<br/>              <mi>x</mi>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mrow>
<br/>          <mo>(</mo>
<br/>          <mrow>
<br/>            <mi>x</mi>
<br/>            <msup>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mi mathvariant="normal">e</mi>
<br/>                </mrow>
<br/>              </mrow>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mo>−<!-- − --></mo>
<br/>                <mi>x</mi>
<br/>              </mrow>
<br/>            </msup>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mfrac>
<br/>                <mrow>
<br/>                  <mrow class="MJX-TeXAtom-ORD">
<br/>                    <mrow class="MJX-TeXAtom-ORD">
<br/>                      <mi mathvariant="normal">d</mi>
<br/>                    </mrow>
<br/>                  </mrow>
<br/>                  <mi>y</mi>
<br/>                </mrow>
<br/>                <mrow>
<br/>                  <mrow class="MJX-TeXAtom-ORD">
<br/>                    <mrow class="MJX-TeXAtom-ORD">
<br/>                      <mi mathvariant="normal">d</mi>
<br/>                    </mrow>
<br/>                  </mrow>
<br/>                  <mi>x</mi>
<br/>                </mrow>
<br/>              </mfrac>
<br/>            </mrow>
<br/>          </mrow>
<br/>          <mo>)</mo>
<br/>        </mrow>
<br/>        <mo>=</mo>
<br/>        <mi>n</mi>
<br/>        <msup>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mi mathvariant="normal">e</mi>
<br/>            </mrow>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>−<!-- − --></mo>
<br/>            <mi>x</mi>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mi>y</mi>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle -{{\\
m {d}} \\\\over {\\
m {d}}x}\\\\left(x{\\
m {e}}^{-x}{{\\
m {d}}y \\\\over {\\
m {d}}x}\\
ight)=n{\\
m {e}}^{-x}y.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83056680ea72a187b53b81b88b52b9feff41ac57" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.125ex; height:6.176ex;" alt="{\\\\displaystyle -{{\\
m {d}} \\\\over {\\
m {d}}x}\\\\left(x{\\
m {e}}^{-x}{{\\
m {d}}y \\\\over {\\
m {d}}x}\\
ight)=n{\\
m {e}}^{-x}y.}"></span></span>
<br/>Cette équation a des solutions non singulières seulement si <span class="texhtml"><i>n</i></span> est un entier positif.
<br/>Les solutions <span class="texhtml"><i>L<sub>n</sub></i></span> forment une suite de polynômes orthogonaux dans <span class="texhtml">L<sup>2</sup></span> (ℝ<sup>+</sup>, <span class="texhtml">e<sup>–<i>x</i></sup>d<i>x</i></span>), et la normalisation se fait en leur imposant d'être de norme 1, donc de former une famille orthonormale. Ils forment même une base hilbertienne de <span class="texhtml">L<sup>2</sup></span>(ℝ<sup>+</sup>, <span class="texhtml">e<sup>–<i>x</i></sup>d<i>x</i></span>). 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Polyn%C3%B4me_de_Laguerre">https://fr.wikipedia.org/wiki/Polyn%C3%B4me_de_Laguerre</a>)"""@fr ;
  skos:prefLabel "Laguerre polynomial"@en, "polynôme de Laguerre"@fr ;
  skos:broader psr:-N2QX9K1Z-L, psr:-VZ83B143-L ;
  skos:inScheme psr: ;
  dc:created "2023-08-16"^^xsd:date ;
  a skos:Concept ;
  dc:modified "2023-08-16"^^xsd:date .

psr: a skos:ConceptScheme .
psr:-N2QX9K1Z-L
  skos:prefLabel "orthogonal polynomials"@en, "polynômes orthogonaux"@fr ;
  a skos:Concept ;
  skos:narrower psr:-DCG8G0KK-F .

psr:-VZ83B143-L
  skos:prefLabel "fonction hypergéométrique"@fr, "hypergeometric function"@en ;
  a skos:Concept ;
  skos:narrower psr:-DCG8G0KK-F .