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http://functions.wolfram.com/06.04.20.0001.01
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D[Multinomial[Subscript[n, 1], Subscript[n, 2], \[Ellipsis],
Subscript[n, m]], Subscript[n, m]] ==
Multinomial[Subscript[n, 1], Subscript[n, 2], \[Ellipsis], Subscript[n, m]]
(PolyGamma[n + 1] - PolyGamma[Subscript[n, m] + 1]) /;
n == Sum[Subscript[n, k], {k, 1, m}]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", SubscriptBox["n", "m"]], RowBox[List["Multinomial", "[", RowBox[List[SubscriptBox["n", "1"], ",", SubscriptBox["n", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["n", "m"]]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List["Multinomial", "[", RowBox[List[SubscriptBox["n", "1"], ",", SubscriptBox["n", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["n", "m"]]], "]"]], RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["n", "+", "1"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[SubscriptBox["n", "m"], "+", "1"]], "]"]]]], ")"]]]]]], "/;", RowBox[List["n", "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", " ", "=", " ", "1"]], RowBox[List[" ", "m"]]], SubscriptBox["n", "k"]]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mfrac> <mrow> <mo> ∂ </mo> <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> … </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> … </mo> <mo> , </mo> <msub> <mi> n </mi> <mi> m </mi> </msub> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["n", "1"], "+", SubscriptBox["n", "2"], "+", "\[Ellipsis]", "+", SubscriptBox["n", "m"]]], ";", SubscriptBox["n", "1"]]], ",", SubscriptBox["n", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["n", "m"]]], ")"]], Multinomial, Rule[Editable, True]] </annotation> </semantics> </mrow> <mrow> <mo> ∂ </mo> <msub> <mi> n </mi> <mi> m </mi> </msub> </mrow> </mfrac> <mo> ⩵ </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> … </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> … </mo> <mo> , </mo> <msub> <mi> n </mi> <mi> m </mi> </msub> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["n", "1"], "+", SubscriptBox["n", "2"], "+", "\[Ellipsis]", "+", SubscriptBox["n", "m"]]], ";", SubscriptBox["n", "1"]]], ",", SubscriptBox["n", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["n", "m"]]], ")"]], Multinomial, Rule[Editable, True]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <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> ⩵ </mo> <mrow> <munderover> <mo> ∑ </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> … </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> m </ci> </apply> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> m </ci> </apply> </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> … </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> m </ci> </apply> </apply> <apply> <plus /> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <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>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[SubscriptBox["n_", "m_"]]]], RowBox[List["Multinomial", "[", RowBox[List[SubscriptBox["n_", "1"], ",", SubscriptBox["n_", "2"], ",", "\[Ellipsis]_", ",", SubscriptBox["n_", "m_"]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["Multinomial", "[", RowBox[List[SubscriptBox["nn", "1"], ",", SubscriptBox["nn", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["nn", "m"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["n", "+", "1"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[SubscriptBox["nn", "m"], "+", "1"]], "]"]]]], ")"]]]], "/;", RowBox[List["n", "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "m"], SubscriptBox["n", "k"]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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