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Multinomial






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Multinomial[n1,n2,...,nm] > Differentiation > Low-order differentiation





http://functions.wolfram.com/06.04.20.0002.01









  


  










Input Form





D[Multinomial[Subscript[n, 1], Subscript[n, 2], \[Ellipsis], Subscript[n, m]], {Subscript[n, m], 2}] == Multinomial[Subscript[n, 1], Subscript[n, 2], \[Ellipsis], Subscript[n, m]] (PolyGamma[Subscript[n, m] + 1]^2 - 2 PolyGamma[1 + Subscript[n, 3]] PolyGamma[n + 1] + PolyGamma[n + 1]^2 - PolyGamma[1, Subscript[n, m] + 1] + PolyGamma[1, n + 1]) /; n == Sum[Subscript[n, k], {k, 1, m}]










Standard Form





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MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mn> 2 </mn> </msup> <semantics> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mo> &#8230; </mo> <mo> + </mo> <msub> <mi> n </mi> <mi> m </mi> </msub> </mrow> <mo> ; </mo> <msub> <mi> n </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> n </mi> <mi> m </mi> </msub> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, RowBox[List[RowBox[List[RowBox[List[SubscriptBox[&quot;n&quot;, &quot;1&quot;], &quot;+&quot;, SubscriptBox[&quot;n&quot;, &quot;2&quot;], &quot;+&quot;, &quot;\[Ellipsis]&quot;, &quot;+&quot;, SubscriptBox[&quot;n&quot;, &quot;m&quot;]]], &quot;;&quot;, SubscriptBox[&quot;n&quot;, &quot;1&quot;]]], &quot;,&quot;, SubscriptBox[&quot;n&quot;, &quot;2&quot;], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, SubscriptBox[&quot;n&quot;, &quot;m&quot;]]], &quot;)&quot;]], Multinomial, Rule[Editable, True]] </annotation> </semantics> </mrow> <mrow> <mo> &#8706; </mo> <msubsup> <mi> n </mi> <mi> m </mi> <mn> 2 </mn> </msubsup> </mrow> </mfrac> <mo> &#10869; </mo> <mrow> <semantics> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <msub> <mi> n </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mo> &#8230; </mo> <mo> + </mo> <msub> <mi> n </mi> <mi> m </mi> </msub> </mrow> <mo> ; </mo> <msub> <mi> n </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <msub> <mi> n </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> n </mi> <mi> m </mi> </msub> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, RowBox[List[RowBox[List[RowBox[List[SubscriptBox[&quot;n&quot;, &quot;1&quot;], &quot;+&quot;, SubscriptBox[&quot;n&quot;, &quot;2&quot;], &quot;+&quot;, &quot;\[Ellipsis]&quot;, &quot;+&quot;, SubscriptBox[&quot;n&quot;, &quot;m&quot;]]], &quot;;&quot;, SubscriptBox[&quot;n&quot;, &quot;1&quot;]]], &quot;,&quot;, SubscriptBox[&quot;n&quot;, &quot;2&quot;], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, SubscriptBox[&quot;n&quot;, &quot;m&quot;]]], &quot;)&quot;]], Multinomial, Rule[Editable, True]] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <msub> <mi> n </mi> <mn> 3 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <msup> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <msub> <mi> n </mi> <mi> m </mi> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <msub> <mi> n </mi> <mi> m </mi> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </munderover> <msub> <mi> n </mi> <mi> k </mi> </msub> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> D </ci> <apply> <ci> Multinomial </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <ci> &#8230; </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> m </ci> </apply> </apply> <list> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </list> </apply> <apply> <times /> <apply> <ci> Multinomial </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <ci> &#8230; </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> m </ci> </apply> </apply> <apply> <plus /> <apply> <power /> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <ci> PolyGamma </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> m </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> m </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <ci> n </ci> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> k </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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Date Added to functions.wolfram.com (modification date)





2001-10-29





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