@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:-SNTKWPJM-D
  skos:prefLabel "polynôme"@fr, "polynomial"@en ;
  a skos:Concept ;
  skos:narrower psr:-ZD1X01FM-W .

psr:-ZD1X01FM-W
  skos:inScheme psr: ;
  skos:definition """A <b>Lommel polynomial</b> <i>R</i><sub><i>m</i>,ν</sub>(<i>z</i>), introduced by Eugen von Lommel&nbsp;(1871), is a polynomial in 1/<i>z</i> giving the recurrence relation 
<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 \\\\displaystyle J_{m+\\
u }(z)=J_{\\
u }(z)R_{m,\\
u }(z)-J_{\\
u -1}(z)R_{m-1,\\
u +1}(z)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mstyle displaystyle="true" scriptlevel="0">
<br/>          <msub>
<br/>            <mi>J</mi>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mi>m</mi>
<br/>              <mo>+</mo>
<br/>              <mi>ν<!-- ν --></mi>
<br/>            </mrow>
<br/>          </msub>
<br/>          <mo stretchy="false">(</mo>
<br/>          <mi>z</mi>
<br/>          <mo stretchy="false">)</mo>
<br/>          <mo>=</mo>
<br/>          <msub>
<br/>            <mi>J</mi>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mi>ν<!-- ν --></mi>
<br/>            </mrow>
<br/>          </msub>
<br/>          <mo stretchy="false">(</mo>
<br/>          <mi>z</mi>
<br/>          <mo stretchy="false">)</mo>
<br/>          <msub>
<br/>            <mi>R</mi>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mi>m</mi>
<br/>              <mo>,</mo>
<br/>              <mi>ν<!-- ν --></mi>
<br/>            </mrow>
<br/>          </msub>
<br/>          <mo stretchy="false">(</mo>
<br/>          <mi>z</mi>
<br/>          <mo stretchy="false">)</mo>
<br/>          <mo>−<!-- − --></mo>
<br/>          <msub>
<br/>            <mi>J</mi>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mi>ν<!-- ν --></mi>
<br/>              <mo>−<!-- − --></mo>
<br/>              <mn>1</mn>
<br/>            </mrow>
<br/>          </msub>
<br/>          <mo stretchy="false">(</mo>
<br/>          <mi>z</mi>
<br/>          <mo stretchy="false">)</mo>
<br/>          <msub>
<br/>            <mi>R</mi>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mi>m</mi>
<br/>              <mo>−<!-- − --></mo>
<br/>              <mn>1</mn>
<br/>              <mo>,</mo>
<br/>              <mi>ν<!-- ν --></mi>
<br/>              <mo>+</mo>
<br/>              <mn>1</mn>
<br/>            </mrow>
<br/>          </msub>
<br/>          <mo stretchy="false">(</mo>
<br/>          <mi>z</mi>
<br/>          <mo stretchy="false">)</mo>
<br/>        </mstyle>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\displaystyle J_{m+\\
u }(z)=J_{\\
u }(z)R_{m,\\
u }(z)-J_{\\
u -1}(z)R_{m-1,\\
u +1}(z)}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae4aafa015cfa644459ff383cd9fde516f955308" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:46.165ex; height:3.009ex;" alt="\\\\displaystyle J_{{m+\\
u }}(z)=J_{\\
u }(z)R_{{m,\\
u }}(z)-J_{{\\
u -1}}(z)R_{{m-1,\\
u +1}}(z)"></span></dd></dl>
<br/>where <i>J</i><sub>ν</sub>(<i>z</i>) is a Bessel function of the first kind.
<br/>They are given explicitly by
<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 R_{m,\\
u }(z)=\\\\sum _{n=0}^{[m/2]}{\\rac {(-1)^{n}(m-n)!\\\\Gamma (\\
u +m-n)}{n!(m-2n)!\\\\Gamma (\\
u +n)}}(z/2)^{2n-m}.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>R</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>m</mi>
<br/>            <mo>,</mo>
<br/>            <mi>ν<!-- ν --></mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>z</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <munderover>
<br/>          <mo>∑<!-- ∑ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>            <mo>=</mo>
<br/>            <mn>0</mn>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo stretchy="false">[</mo>
<br/>            <mi>m</mi>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mo>/</mo>
<br/>            </mrow>
<br/>            <mn>2</mn>
<br/>            <mo stretchy="false">]</mo>
<br/>          </mrow>
<br/>        </munderover>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mo>−<!-- − --></mo>
<br/>              <mn>1</mn>
<br/>              <msup>
<br/>                <mo stretchy="false">)</mo>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mi>n</mi>
<br/>                </mrow>
<br/>              </msup>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>m</mi>
<br/>              <mo>−<!-- − --></mo>
<br/>              <mi>n</mi>
<br/>              <mo stretchy="false">)</mo>
<br/>              <mo>!</mo>
<br/>              <mi mathvariant="normal">Γ<!-- Γ --></mi>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>ν<!-- ν --></mi>
<br/>              <mo>+</mo>
<br/>              <mi>m</mi>
<br/>              <mo>−<!-- − --></mo>
<br/>              <mi>n</mi>
<br/>              <mo stretchy="false">)</mo>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <mi>n</mi>
<br/>              <mo>!</mo>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>m</mi>
<br/>              <mo>−<!-- − --></mo>
<br/>              <mn>2</mn>
<br/>              <mi>n</mi>
<br/>              <mo stretchy="false">)</mo>
<br/>              <mo>!</mo>
<br/>              <mi mathvariant="normal">Γ<!-- Γ --></mi>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>ν<!-- ν --></mi>
<br/>              <mo>+</mo>
<br/>              <mi>n</mi>
<br/>              <mo stretchy="false">)</mo>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>z</mi>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mo>/</mo>
<br/>        </mrow>
<br/>        <mn>2</mn>
<br/>        <msup>
<br/>          <mo stretchy="false">)</mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>2</mn>
<br/>            <mi>n</mi>
<br/>            <mo>−<!-- − --></mo>
<br/>            <mi>m</mi>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle R_{m,\\
u }(z)=\\\\sum _{n=0}^{[m/2]}{\\rac {(-1)^{n}(m-n)!\\\\Gamma (\\
u +m-n)}{n!(m-2n)!\\\\Gamma (\\
u +n)}}(z/2)^{2n-m}.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c01d46ac1e0813de47d962ee9ba94be163df843a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.005ex; width:54.959ex; height:7.676ex;" alt="{\\\\displaystyle R_{m,\\
u }(z)=\\\\sum _{n=0}^{[m/2]}{\\rac {(-1)^{n}(m-n)!\\\\Gamma (\\
u +m-n)}{n!(m-2n)!\\\\Gamma (\\
u +n)}}(z/2)^{2n-m}.}"> 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Lommel_polynomial">https://en.wikipedia.org/wiki/Lommel_polynomial</a>)"""@en, """Les <b>polynômes de Lommel</b>, <i>R</i><sub><i>m</i>,ν</sub>(<i>z</i>), introduits par Eugen von Lommel en 1871, sont des polynômes en 1/<i>z</i> vérifiant la relation suivante:
<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 \\\\displaystyle J_{m+\\
u }(z)=J_{\\
u }(z)R_{m,\\
u }(z)-J_{\\
u -1}(z)R_{m-1,\\
u +1}(z)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mstyle displaystyle="true" scriptlevel="0">
<br/>          <msub>
<br/>            <mi>J</mi>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mi>m</mi>
<br/>              <mo>+</mo>
<br/>              <mi>ν<!-- ν --></mi>
<br/>            </mrow>
<br/>          </msub>
<br/>          <mo stretchy="false">(</mo>
<br/>          <mi>z</mi>
<br/>          <mo stretchy="false">)</mo>
<br/>          <mo>=</mo>
<br/>          <msub>
<br/>            <mi>J</mi>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mi>ν<!-- ν --></mi>
<br/>            </mrow>
<br/>          </msub>
<br/>          <mo stretchy="false">(</mo>
<br/>          <mi>z</mi>
<br/>          <mo stretchy="false">)</mo>
<br/>          <msub>
<br/>            <mi>R</mi>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mi>m</mi>
<br/>              <mo>,</mo>
<br/>              <mi>ν<!-- ν --></mi>
<br/>            </mrow>
<br/>          </msub>
<br/>          <mo stretchy="false">(</mo>
<br/>          <mi>z</mi>
<br/>          <mo stretchy="false">)</mo>
<br/>          <mo>−<!-- − --></mo>
<br/>          <msub>
<br/>            <mi>J</mi>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mi>ν<!-- ν --></mi>
<br/>              <mo>−<!-- − --></mo>
<br/>              <mn>1</mn>
<br/>            </mrow>
<br/>          </msub>
<br/>          <mo stretchy="false">(</mo>
<br/>          <mi>z</mi>
<br/>          <mo stretchy="false">)</mo>
<br/>          <msub>
<br/>            <mi>R</mi>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mi>m</mi>
<br/>              <mo>−<!-- − --></mo>
<br/>              <mn>1</mn>
<br/>              <mo>,</mo>
<br/>              <mi>ν<!-- ν --></mi>
<br/>              <mo>+</mo>
<br/>              <mn>1</mn>
<br/>            </mrow>
<br/>          </msub>
<br/>          <mo stretchy="false">(</mo>
<br/>          <mi>z</mi>
<br/>          <mo stretchy="false">)</mo>
<br/>        </mstyle>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\displaystyle J_{m+\\
u }(z)=J_{\\
u }(z)R_{m,\\
u }(z)-J_{\\
u -1}(z)R_{m-1,\\
u +1}(z)}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae4aafa015cfa644459ff383cd9fde516f955308" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:46.165ex; height:3.009ex;" alt="{\\\\displaystyle \\\\displaystyle J_{m+\\
u }(z)=J_{\\
u }(z)R_{m,\\
u }(z)-J_{\\
u -1}(z)R_{m-1,\\
u +1}(z)}"></span></dd></dl>
<br/>où <i>J</i><sub>ν</sub>(<i>z</i>) est la fonction de Bessel du premier ordre.
<br/>Ils sont donnés explicitement par
<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 R_{m,\\
u }=\\\\sum _{n=0}^{[m/2]}{\\rac {(-1)^{m}(m-n)!\\\\Gamma (\\
u +m-n)}{n!(m-2n)!\\\\Gamma (\\
u +n)}}(z/2)^{2n-m}.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>R</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>m</mi>
<br/>            <mo>,</mo>
<br/>            <mi>ν<!-- ν --></mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>=</mo>
<br/>        <munderover>
<br/>          <mo>∑<!-- ∑ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>            <mo>=</mo>
<br/>            <mn>0</mn>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo stretchy="false">[</mo>
<br/>            <mi>m</mi>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mo>/</mo>
<br/>            </mrow>
<br/>            <mn>2</mn>
<br/>            <mo stretchy="false">]</mo>
<br/>          </mrow>
<br/>        </munderover>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mo>−<!-- − --></mo>
<br/>              <mn>1</mn>
<br/>              <msup>
<br/>                <mo stretchy="false">)</mo>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mi>m</mi>
<br/>                </mrow>
<br/>              </msup>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>m</mi>
<br/>              <mo>−<!-- − --></mo>
<br/>              <mi>n</mi>
<br/>              <mo stretchy="false">)</mo>
<br/>              <mo>!</mo>
<br/>              <mi mathvariant="normal">Γ<!-- Γ --></mi>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>ν<!-- ν --></mi>
<br/>              <mo>+</mo>
<br/>              <mi>m</mi>
<br/>              <mo>−<!-- − --></mo>
<br/>              <mi>n</mi>
<br/>              <mo stretchy="false">)</mo>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <mi>n</mi>
<br/>              <mo>!</mo>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>m</mi>
<br/>              <mo>−<!-- − --></mo>
<br/>              <mn>2</mn>
<br/>              <mi>n</mi>
<br/>              <mo stretchy="false">)</mo>
<br/>              <mo>!</mo>
<br/>              <mi mathvariant="normal">Γ<!-- Γ --></mi>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>ν<!-- ν --></mi>
<br/>              <mo>+</mo>
<br/>              <mi>n</mi>
<br/>              <mo stretchy="false">)</mo>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>z</mi>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mo>/</mo>
<br/>        </mrow>
<br/>        <mn>2</mn>
<br/>        <msup>
<br/>          <mo stretchy="false">)</mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>2</mn>
<br/>            <mi>n</mi>
<br/>            <mo>−<!-- − --></mo>
<br/>            <mi>m</mi>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle R_{m,\\
u }=\\\\sum _{n=0}^{[m/2]}{\\rac {(-1)^{m}(m-n)!\\\\Gamma (\\
u +m-n)}{n!(m-2n)!\\\\Gamma (\\
u +n)}}(z/2)^{2n-m}.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8816aaf67f8b2c54973f0f0de421588ea8aedfd6" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.005ex; width:52.519ex; height:7.676ex;" alt="{\\\\displaystyle R_{m,\\
u }=\\\\sum _{n=0}^{[m/2]}{\\rac {(-1)^{m}(m-n)!\\\\Gamma (\\
u +m-n)}{n!(m-2n)!\\\\Gamma (\\
u +n)}}(z/2)^{2n-m}.}"></span></dd></dl>
<br/>où Γ désigne la fonction gamma.
<br/>Ils sont utilisés en tant qu'outil de démonstration en théorie de la transcendance. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Polyn%C3%B4me_de_Lommel">https://fr.wikipedia.org/wiki/Polyn%C3%B4me_de_Lommel</a>)"""@fr ;
  dc:modified "2023-08-16"^^xsd:date ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Lommel_polynomial>, <https://fr.wikipedia.org/wiki/Polyn%C3%B4me_de_Lommel> ;
  a skos:Concept ;
  dc:created "2023-08-16"^^xsd:date ;
  skos:prefLabel "polynôme de Lommel"@fr, "Lommel polynomial"@en ;
  skos:broader psr:-SNTKWPJM-D, psr:-FH1H1FB9-1 .

psr:-FH1H1FB9-1
  skos:prefLabel "special function"@en, "fonction spéciale"@fr ;
  a skos:Concept ;
  skos:narrower psr:-ZD1X01FM-W .