@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:-VZ83B143-L
  skos:prefLabel "fonction hypergéométrique"@fr, "hypergeometric function"@en ;
  a skos:Concept ;
  skos:narrower psr:-HDF2GXJ6-4 .

psr:-WTK9N44N-V
  skos:prefLabel "inégalité d'Askey-Gasper"@fr, "Askey-Gasper inequality"@en ;
  a skos:Concept ;
  skos:related psr:-HDF2GXJ6-4 .

psr:-HZ8HW1CT-5
  skos:prefLabel "hypergeometric series"@en, "série hypergéométrique"@fr ;
  a skos:Concept ;
  skos:related psr:-HDF2GXJ6-4 .

psr:-N2QX9K1Z-L
  skos:prefLabel "orthogonal polynomials"@en, "polynômes orthogonaux"@fr ;
  a skos:Concept ;
  skos:narrower psr:-HDF2GXJ6-4 .

psr: a skos:ConceptScheme .
psr:-HDF2GXJ6-4
  skos:prefLabel "polynôme de Jacobi"@fr, "Jacobi polynomial"@en ;
  skos:definition """En mathématiques, les <b>polynômes de Jacobi</b> sont une classe de polynômes orthogonaux. 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 P_{n}^{(\\\\alpha ,\\eta )}(z)={\\rac {(\\\\alpha +1)_{n}}{n!}}\\\\,_{2}F_{1}\\\\left(-n,1+\\\\alpha +\\eta +n;\\\\alpha +1;{\\rac {1-z}{2}}\\
ight),}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msubsup>
<br/>          <mi>P</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>,</mo>
<br/>            <mi>β<!-- β --></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/>              <mi>α<!-- α --></mi>
<br/>              <mo>+</mo>
<br/>              <mn>1</mn>
<br/>              <msub>
<br/>                <mo stretchy="false">)</mo>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mi>n</mi>
<br/>                </mrow>
<br/>              </msub>
<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>1</mn>
<br/>            <mo>+</mo>
<br/>            <mi>α<!-- α --></mi>
<br/>            <mo>+</mo>
<br/>            <mi>β<!-- β --></mi>
<br/>            <mo>+</mo>
<br/>            <mi>n</mi>
<br/>            <mo>;</mo>
<br/>            <mi>α<!-- α --></mi>
<br/>            <mo>+</mo>
<br/>            <mn>1</mn>
<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/>        <mo>,</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle P_{n}^{(\\\\alpha ,\\eta )}(z)={\\rac {(\\\\alpha +1)_{n}}{n!}}\\\\,_{2}F_{1}\\\\left(-n,1+\\\\alpha +\\eta +n;\\\\alpha +1;{\\rac {1-z}{2}}\\
ight),}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0e9e51a17de3666a18693c705746a76063a6581" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:61.587ex; height:6.343ex;" alt="{\\\\displaystyle P_{n}^{(\\\\alpha ,\\eta )}(z)={\\rac {(\\\\alpha +1)_{n}}{n!}}\\\\,_{2}F_{1}\\\\left(-n,1+\\\\alpha +\\eta +n;\\\\alpha +1;{\\rac {1-z}{2}}\\
ight),}"></span></dd></dl>
<br/>où <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 (\\\\alpha +1)_{n}\\\\,}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>α<!-- α --></mi>
<br/>        <mo>+</mo>
<br/>        <mn>1</mn>
<br/>        <msub>
<br/>          <mo stretchy="false">)</mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle (\\\\alpha +1)_{n}\\\\,}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62912ad94a42319322a4a9935df8c387c3784496" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.905ex; height:2.843ex;" alt="{\\\\displaystyle (\\\\alpha +1)_{n}\\\\,}"></span> est le symbole de Pochhammer pour la factorielle croissante, (Abramowitz &amp; Stegun p561.) et ainsi, nous avons l'expression explicite
<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 P_{n}^{(\\\\alpha ,\\eta )}(z)={\\rac {\\\\Gamma (\\\\alpha +n+1)}{n!\\\\Gamma (\\\\alpha +\\eta +n+1)}}\\\\sum _{m=0}^{n}{n \\\\choose m}{\\rac {\\\\Gamma (\\\\alpha +\\eta +n+m+1)}{\\\\Gamma (\\\\alpha +m+1)}}\\\\left({\\rac {z-1}{2}}\\
ight)^{m},}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msubsup>
<br/>          <mi>P</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>,</mo>
<br/>            <mi>β<!-- β --></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/>              <mi mathvariant="normal">Γ<!-- Γ --></mi>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>α<!-- α --></mi>
<br/>              <mo>+</mo>
<br/>              <mi>n</mi>
<br/>              <mo>+</mo>
<br/>              <mn>1</mn>
<br/>              <mo stretchy="false">)</mo>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <mi>n</mi>
<br/>              <mo>!</mo>
<br/>              <mi mathvariant="normal">Γ<!-- Γ --></mi>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>α<!-- α --></mi>
<br/>              <mo>+</mo>
<br/>              <mi>β<!-- β --></mi>
<br/>              <mo>+</mo>
<br/>              <mi>n</mi>
<br/>              <mo>+</mo>
<br/>              <mn>1</mn>
<br/>              <mo stretchy="false">)</mo>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <munderover>
<br/>          <mo>∑<!-- ∑ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>m</mi>
<br/>            <mo>=</mo>
<br/>            <mn>0</mn>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </munderover>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow>
<br/>            <mrow class="MJX-TeXAtom-OPEN">
<br/>              <mo maxsize="2.047em" minsize="2.047em">(</mo>
<br/>            </mrow>
<br/>            <mfrac linethickness="0">
<br/>              <mi>n</mi>
<br/>              <mi>m</mi>
<br/>            </mfrac>
<br/>            <mrow class="MJX-TeXAtom-CLOSE">
<br/>              <mo maxsize="2.047em" minsize="2.047em">)</mo>
<br/>            </mrow>
<br/>          </mrow>
<br/>        </mrow>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <mi mathvariant="normal">Γ<!-- Γ --></mi>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>α<!-- α --></mi>
<br/>              <mo>+</mo>
<br/>              <mi>β<!-- β --></mi>
<br/>              <mo>+</mo>
<br/>              <mi>n</mi>
<br/>              <mo>+</mo>
<br/>              <mi>m</mi>
<br/>              <mo>+</mo>
<br/>              <mn>1</mn>
<br/>              <mo stretchy="false">)</mo>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <mi mathvariant="normal">Γ<!-- Γ --></mi>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>α<!-- α --></mi>
<br/>              <mo>+</mo>
<br/>              <mi>m</mi>
<br/>              <mo>+</mo>
<br/>              <mn>1</mn>
<br/>              <mo stretchy="false">)</mo>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <msup>
<br/>          <mrow>
<br/>            <mo>(</mo>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mfrac>
<br/>                <mrow>
<br/>                  <mi>z</mi>
<br/>                  <mo>−<!-- − --></mo>
<br/>                  <mn>1</mn>
<br/>                </mrow>
<br/>                <mn>2</mn>
<br/>              </mfrac>
<br/>            </mrow>
<br/>            <mo>)</mo>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>m</mi>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>,</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle P_{n}^{(\\\\alpha ,\\eta )}(z)={\\rac {\\\\Gamma (\\\\alpha +n+1)}{n!\\\\Gamma (\\\\alpha +\\eta +n+1)}}\\\\sum _{m=0}^{n}{n \\\\choose m}{\\rac {\\\\Gamma (\\\\alpha +\\eta +n+m+1)}{\\\\Gamma (\\\\alpha +m+1)}}\\\\left({\\rac {z-1}{2}}\\
ight)^{m},}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/962e36fdbe757daeefe6c4da13df6daab5dbc8be" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:76.144ex; height:6.843ex;" alt="{\\\\displaystyle P_{n}^{(\\\\alpha ,\\eta )}(z)={\\rac {\\\\Gamma (\\\\alpha +n+1)}{n!\\\\Gamma (\\\\alpha +\\eta +n+1)}}\\\\sum _{m=0}^{n}{n \\\\choose m}{\\rac {\\\\Gamma (\\\\alpha +\\eta +n+m+1)}{\\\\Gamma (\\\\alpha +m+1)}}\\\\left({\\rac {z-1}{2}}\\
ight)^{m},}"></span></dd></dl>
<br/>pour laquelle la valeur finale est
<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 P_{n}^{(\\\\alpha ,\\eta )}(1)={n+\\\\alpha  \\\\choose n}.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msubsup>
<br/>          <mi>P</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>,</mo>
<br/>            <mi>β<!-- β --></mi>
<br/>            <mo stretchy="false">)</mo>
<br/>          </mrow>
<br/>        </msubsup>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mn>1</mn>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow>
<br/>            <mrow class="MJX-TeXAtom-OPEN">
<br/>              <mo maxsize="2.047em" minsize="2.047em">(</mo>
<br/>            </mrow>
<br/>            <mfrac linethickness="0">
<br/>              <mrow>
<br/>                <mi>n</mi>
<br/>                <mo>+</mo>
<br/>                <mi>α<!-- α --></mi>
<br/>              </mrow>
<br/>              <mi>n</mi>
<br/>            </mfrac>
<br/>            <mrow class="MJX-TeXAtom-CLOSE">
<br/>              <mo maxsize="2.047em" minsize="2.047em">)</mo>
<br/>            </mrow>
<br/>          </mrow>
<br/>        </mrow>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle P_{n}^{(\\\\alpha ,\\eta )}(1)={n+\\\\alpha  \\\\choose n}.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ea914c1a25729ba439473ae730691436177d86e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.645ex; height:6.176ex;" alt="{\\\\displaystyle P_{n}^{(\\\\alpha ,\\eta )}(1)={n+\\\\alpha  \\\\choose n}.}"> 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Polyn%C3%B4me_de_Jacobi">https://fr.wikipedia.org/wiki/Polyn%C3%B4me_de_Jacobi</a>)"""@fr, """In mathematics, <b>Jacobi polynomials</b> (occasionally called <b>hypergeometric polynomials</b>) <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 P_{n}^{(\\\\alpha ,\\eta )}(x)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msubsup>
<br/>          <mi>P</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>,</mo>
<br/>            <mi>β<!-- β --></mi>
<br/>            <mo stretchy="false">)</mo>
<br/>          </mrow>
<br/>        </msubsup>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle P_{n}^{(\\\\alpha ,\\eta )}(x)}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2594fb2b13259616c95748f0eb78c63b5711f979" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.923ex; height:3.509ex;" alt="{\\\\displaystyle P_{n}^{(\\\\alpha ,\\eta )}(x)}"></span>
<br/>are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight
<br/><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 (1-x)^{\\\\alpha }(1+x)^{\\eta }}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mo stretchy="false">(</mo>
<br/>        <mn>1</mn>
<br/>        <mo>−<!-- − --></mo>
<br/>        <mi>x</mi>
<br/>        <msup>
<br/>          <mo stretchy="false">)</mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>α<!-- α --></mi>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mn>1</mn>
<br/>        <mo>+</mo>
<br/>        <mi>x</mi>
<br/>        <msup>
<br/>          <mo stretchy="false">)</mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>β<!-- β --></mi>
<br/>          </mrow>
<br/>        </msup>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle (1-x)^{\\\\alpha }(1+x)^{\\eta }}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/095d74821e4cb35a12ab9e9e2044037b52811ed5" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.742ex; height:3.176ex;" alt="{\\\\displaystyle (1-x)^{\\\\alpha }(1+x)^{\\eta }}"></span> on the interval <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 [-1,1]}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mo stretchy="false">[</mo>
<br/>        <mo>−<!-- − --></mo>
<br/>        <mn>1</mn>
<br/>        <mo>,</mo>
<br/>        <mn>1</mn>
<br/>        <mo stretchy="false">]</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle [-1,1]}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51e3b7f14a6f70e614728c583409a0b9a8b9de01" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.461ex; height:2.843ex;" alt="[-1,1]"></span>. The Gegenbauer polynomials, and thus also the Legendre, Zernike and Chebyshev polynomials, are special cases of the Jacobi polynomials. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Jacobi_polynomials">https://en.wikipedia.org/wiki/Jacobi_polynomials</a>)"""@en ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Jacobi_polynomials>, <https://fr.wikipedia.org/wiki/Polyn%C3%B4me_de_Jacobi> ;
  skos:altLabel "hypergeometric polynomial"@en ;
  skos:inScheme psr: ;
  skos:broader psr:-VZ83B143-L, psr:-N2QX9K1Z-L ;
  dc:modified "2023-08-16"^^xsd:date ;
  a skos:Concept ;
  skos:related psr:-WTK9N44N-V, psr:-HZ8HW1CT-5 .