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Multinomial






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Multinomial[n1,n2,...,nm] > Differentiation > Symbolic differentiation





http://functions.wolfram.com/06.04.20.0003.02









  


  










Input Form





D[Multinomial[Subscript[n, 1], Subscript[n, 2], \[Ellipsis], Subscript[n, m]], {Subscript[n, m], u}] == (((-1)^u u! Gamma[s + 1]^(u + 1))/Product[Gamma[Subscript[n, k] + 1] Gamma[Subscript[n, m] - s], {k, 1, m - 1}]) HypergeometricPFQRegularized[{Subscript[a, 1], Subscript[a, 2], \[Ellipsis], Subscript[a, u + 1], s - Subscript[n, m] + 1}, {1 + Subscript[a, 1], 1 + Subscript[a, 2], \[Ellipsis], 1 + Subscript[a, u + 1]}, 1] /; Subscript[a, 1] == Subscript[a, 2] == \[Ellipsis] == Subscript[a, u + 1] == s + 1 && s == Sum[Subscript[n, k], {k, 1, m}] && Element[u, Integers] && u >= 0 && !(Element[s - Subscript[n, m], Integers] && s - Subscript[n, m] >= 0)










Standard Form





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MathML Form







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</ci> <apply> <ci> Subscript </ci> <ci> a </ci> <apply> <plus /> <ci> u </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <plus /> <ci> s </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> m </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </list> <list> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <ci> &#8230; </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <apply> <plus /> <ci> u </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </list> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <ci> &#8230; 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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["n_", "m_"], ",", "u_"]], "}"]]]]], RowBox[List["Multinomial", "[", RowBox[List[SubscriptBox["n_", "1"], ",", SubscriptBox["n_", "2"], ",", "\[Ellipsis]_", ",", SubscriptBox["n_", "m_"]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "u"], " ", RowBox[List["u", "!"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["s", "+", "1"]], "]"]], RowBox[List["u", "+", "1"]]]]], ")"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["u", "+", "1"]]], ",", RowBox[List["s", "-", SubscriptBox["nn", "m"], "+", "1"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "1"]]], ",", RowBox[List["1", "+", SubscriptBox["a", "2"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["a", RowBox[List["u", "+", "1"]]]]]]], "}"]], ",", "1"]], "]"]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], RowBox[List["m", "-", "1"]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["nn", "k"], "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["nn", "m"], "-", "s"]], "]"]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", "1"], "\[Equal]", SubscriptBox["a", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["a", RowBox[List["u", "+", "1"]]], "\[Equal]", RowBox[List["s", "+", "1"]]]], "&&", RowBox[List["s", "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "m"], SubscriptBox["nn", "k"]]]]], "&&", RowBox[List["u", "\[Element]", "Integers"]], "&&", RowBox[List["u", "\[GreaterEqual]", "0"]], "&&", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["s", "-", SubscriptBox["nn", "m"]]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["s", "-", SubscriptBox["nn", "m"]]], "\[GreaterEqual]", "0"]]]], ")"]]]]]]]]]]]]










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





2001-10-29





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