@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: a skos:ConceptScheme .
psr:-FC602064-K
  skos:inScheme psr: ;
  dc:modified "2023-08-16"^^xsd:date ;
  skos:altLabel "polynôme ultrasphérique"@fr, "ultraspherical polynomial"@en ;
  a skos:Concept ;
  skos:definition """En mathématiques, les <b>polynômes de Gegenbauer</b> ou <b>polynômes ultrasphériques</b> sont une classe de polynômes orthogonaux. Ils sont nommés ainsi en l'honneur de Leopold Gegenbauer (1849-1903). Ils sont obtenus à partir des séries hypergéométriques dans les cas où la série est en fait finie&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 C_{n}^{(\\\\alpha )}(z)={\\rac {(2\\\\alpha )^{\\\\underline {n}}}{n!}}\\\\,_{2}F_{1}\\\\left(-n,2\\\\alpha +n;\\\\alpha +{\\rac {1}{2}};{\\rac {1-z}{2}}\\
ight)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msubsup>
<br/>          <mi>C</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo stretchy="false">(</mo>
<br/>            <mi>α<!-- α --></mi>
<br/>            <mo stretchy="false">)</mo>
<br/>          </mrow>
<br/>        </msubsup>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>z</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mn>2</mn>
<br/>              <mi>α<!-- α --></mi>
<br/>              <msup>
<br/>                <mo stretchy="false">)</mo>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <munder>
<br/>                    <mi>n</mi>
<br/>                    <mo>_<!-- _ --></mo>
<br/>                  </munder>
<br/>                </mrow>
<br/>              </msup>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <mi>n</mi>
<br/>              <mo>!</mo>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <msub>
<br/>          <mspace width="thinmathspace"></mspace>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>2</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>        <msub>
<br/>          <mi>F</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mrow>
<br/>          <mo>(</mo>
<br/>          <mrow>
<br/>            <mo>−<!-- − --></mo>
<br/>            <mi>n</mi>
<br/>            <mo>,</mo>
<br/>            <mn>2</mn>
<br/>            <mi>α<!-- α --></mi>
<br/>            <mo>+</mo>
<br/>            <mi>n</mi>
<br/>            <mo>;</mo>
<br/>            <mi>α<!-- α --></mi>
<br/>            <mo>+</mo>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mfrac>
<br/>                <mn>1</mn>
<br/>                <mn>2</mn>
<br/>              </mfrac>
<br/>            </mrow>
<br/>            <mo>;</mo>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mfrac>
<br/>                <mrow>
<br/>                  <mn>1</mn>
<br/>                  <mo>−<!-- − --></mo>
<br/>                  <mi>z</mi>
<br/>                </mrow>
<br/>                <mn>2</mn>
<br/>              </mfrac>
<br/>            </mrow>
<br/>          </mrow>
<br/>          <mo>)</mo>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle C_{n}^{(\\\\alpha )}(z)={\\rac {(2\\\\alpha )^{\\\\underline {n}}}{n!}}\\\\,_{2}F_{1}\\\\left(-n,2\\\\alpha +n;\\\\alpha +{\\rac {1}{2}};{\\rac {1-z}{2}}\\
ight)}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/264bd3ea88025faa0b2c5fb1cfcc7a46cfb658bf" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:50.115ex; height:6.343ex;" alt="{\\\\displaystyle C_{n}^{(\\\\alpha )}(z)={\\rac {(2\\\\alpha )^{\\\\underline {n}}}{n!}}\\\\,_{2}F_{1}\\\\left(-n,2\\\\alpha +n;\\\\alpha +{\\rac {1}{2}};{\\rac {1-z}{2}}\\
ight)}"></span></dd></dl>
<br/>où <span style="text-decoration: underline;"><span class="texhtml mvar" style="font-style:italic;">n</span></span> est la factorielle décroissante 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Polyn%C3%B4me_de_Gegenbauer">https://fr.wikipedia.org/wiki/Polyn%C3%B4me_de_Gegenbauer</a>)"""@fr, """In mathematics, <b>Gegenbauer polynomials</b> or <b>ultraspherical polynomials</b> <i>C</i><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">(α)</sup><br><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"><i>n</i></sub></span>(<i>x</i>) are  orthogonal polynomials on the interval [−1,1] with respect to the weight function (1&nbsp;−&nbsp;<i>x</i><sup>2</sup>)<sup><i>α</i>–1/2</sup>. They generalize Legendre polynomials and Chebyshev polynomials, and are special cases of Jacobi polynomials. They are named after Leopold Gegenbauer. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Gegenbauer_polynomials">https://en.wikipedia.org/wiki/Gegenbauer_polynomials</a>)"""@en ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Gegenbauer_polynomials>, <https://fr.wikipedia.org/wiki/Polyn%C3%B4me_de_Gegenbauer> ;
  dc:created "2023-08-16"^^xsd:date ;
  skos:broader psr:-VZ83B143-L, psr:-N2QX9K1Z-L ;
  skos:prefLabel "Gegenbauer polynomial"@en, "polynôme de Gegenbauer"@fr .

psr:-N2QX9K1Z-L
  skos:prefLabel "orthogonal polynomials"@en, "polynômes orthogonaux"@fr ;
  a skos:Concept ;
  skos:narrower psr:-FC602064-K .

psr:-VZ83B143-L
  skos:prefLabel "fonction hypergéométrique"@fr, "hypergeometric function"@en ;
  a skos:Concept ;
  skos:narrower psr:-FC602064-K .