@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:-K9FXDR6F-N
  skos:prefLabel "loi de probabilité"@fr, "probability distribution"@en ;
  a skos:Concept ;
  skos:narrower psr:-P7HZW9R6-S .

psr:-P7HZW9R6-S
  a skos:Concept ;
  skos:altLabel "Pearson-Cunningham function"@en ;
  dc:created "2023-08-17"^^xsd:date ;
  skos:broader psr:-VZ83B143-L, psr:-K9FXDR6F-N ;
  skos:prefLabel "Cunningham function"@en, "fonction de Cunningham"@fr ;
  dc:modified "2023-08-23"^^xsd:date ;
  skos:inScheme psr: ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Cunningham_function> ;
  skos:definition """In statistics, the <b>Cunningham function</b> or <b>Pearson–Cunningham function</b> ω<sub><i>m</i>,<i>n</i></sub>(<i>x</i>) is a generalisation of a special function introduced by Pearson (1906) and studied in the form here by Cunningham (1908). It can be defined in terms of  the confluent hypergeometric function <i>U</i>, 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 \\\\displaystyle \\\\omega _{m,n}(x)={\\rac {e^{-x+\\\\pi i(m/2-n)}}{\\\\Gamma (1+n-m/2)}}U(m/2-n,1+m,x).}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mstyle displaystyle="true" scriptlevel="0">
<br/>          <msub>
<br/>            <mi>ω<!-- ω --></mi>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mi>m</mi>
<br/>              <mo>,</mo>
<br/>              <mi>n</mi>
<br/>            </mrow>
<br/>          </msub>
<br/>          <mo stretchy="false">(</mo>
<br/>          <mi>x</mi>
<br/>          <mo stretchy="false">)</mo>
<br/>          <mo>=</mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mfrac>
<br/>              <msup>
<br/>                <mi>e</mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mo>−<!-- − --></mo>
<br/>                  <mi>x</mi>
<br/>                  <mo>+</mo>
<br/>                  <mi>π<!-- π --></mi>
<br/>                  <mi>i</mi>
<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>−<!-- − --></mo>
<br/>                  <mi>n</mi>
<br/>                  <mo stretchy="false">)</mo>
<br/>                </mrow>
<br/>              </msup>
<br/>              <mrow>
<br/>                <mi mathvariant="normal">Γ<!-- Γ --></mi>
<br/>                <mo stretchy="false">(</mo>
<br/>                <mn>1</mn>
<br/>                <mo>+</mo>
<br/>                <mi>n</mi>
<br/>                <mo>−<!-- − --></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/>            </mfrac>
<br/>          </mrow>
<br/>          <mi>U</mi>
<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>−<!-- − --></mo>
<br/>          <mi>n</mi>
<br/>          <mo>,</mo>
<br/>          <mn>1</mn>
<br/>          <mo>+</mo>
<br/>          <mi>m</mi>
<br/>          <mo>,</mo>
<br/>          <mi>x</mi>
<br/>          <mo stretchy="false">)</mo>
<br/>          <mo>.</mo>
<br/>        </mstyle>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\displaystyle \\\\omega _{m,n}(x)={\\rac {e^{-x+\\\\pi i(m/2-n)}}{\\\\Gamma (1+n-m/2)}}U(m/2-n,1+m,x).}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29677b0c3c71cb0bddb321ba06f4c3a193612374" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.671ex; width:49.783ex; height:6.509ex;" alt="\\\\displaystyle \\\\omega _{{m,n}}(x)={\\rac  {e^{{-x+\\\\pi i(m/2-n)}}}{\\\\Gamma (1+n-m/2)}}U(m/2-n,1+m,x)."></span></dd></dl>
<br/>The function was studied by Cunningham in the context of a multivariate generalisation of the Edgeworth expansion for approximating a probability density function based on its (joint) moments. In a more general context, the function is related to the solution of the constant-coefficient diffusion equation, in one or more dimensions.  
<br/>The function ω<sub><i>m</i>,<i>n</i></sub>(<i>x</i>) is a solution of the differential equation for <i>X</i>:
<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 xX''+(x+1+m)X'+(n+{\\	frac {1}{2}}m+1)X.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>x</mi>
<br/>        <msup>
<br/>          <mi>X</mi>
<br/>          <mo>″</mo>
<br/>        </msup>
<br/>        <mo>+</mo>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo>+</mo>
<br/>        <mn>1</mn>
<br/>        <mo>+</mo>
<br/>        <mi>m</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <msup>
<br/>          <mi>X</mi>
<br/>          <mo>′</mo>
<br/>        </msup>
<br/>        <mo>+</mo>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>n</mi>
<br/>        <mo>+</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mstyle displaystyle="false" scriptlevel="0">
<br/>            <mfrac>
<br/>              <mn>1</mn>
<br/>              <mn>2</mn>
<br/>            </mfrac>
<br/>          </mstyle>
<br/>        </mrow>
<br/>        <mi>m</mi>
<br/>        <mo>+</mo>
<br/>        <mn>1</mn>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mi>X</mi>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle xX''+(x+1+m)X'+(n+{\\	frac {1}{2}}m+1)X.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c127df020e79d82fd95ba7879e425002a1fb6473" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.171ex; width:41.221ex; height:3.509ex;" alt="xX''+(x+1+m)X'+(n+{\\	frac  {1}{2}}m+1)X."></span></dd></dl>
<br/>The special function studied by Pearson is given, in his notation 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 \\\\omega _{2n}(x)=\\\\omega _{0,n}(x).}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>ω<!-- ω --></mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>2</mn>
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <msub>
<br/>          <mi>ω<!-- ω --></mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>0</mn>
<br/>            <mo>,</mo>
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\omega _{2n}(x)=\\\\omega _{0,n}(x).}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/615e4a8b7d368c1fa76f837e9d3cf9d56c66e491" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:17.453ex; height:3.009ex;" alt="\\\\omega _{{2n}}(x)=\\\\omega _{{0,n}}(x)."> 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Cunningham_function">https://en.wikipedia.org/wiki/Cunningham_function</a>)"""@en .

psr:-VZ83B143-L
  skos:prefLabel "fonction hypergéométrique"@fr, "hypergeometric function"@en ;
  a skos:Concept ;
  skos:narrower psr:-P7HZW9R6-S .