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 Multinomial

 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

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List[SubscriptBox["n", "m"], ",", "u"]], "}"]]], RowBox[List["Multinomial", "[", RowBox[List[SubscriptBox["n", "1"], ",", SubscriptBox["n", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["n", "m"]]], "]"]]]], "\[Equal]", RowBox[List[FractionBox[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[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], RowBox[List["m", "-", "1"]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["n", "k"], "+", "1"]], "]"]], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["n", "m"], "-", "s"]], "]"]]]]]]], 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["n", "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[RowBox[List[SubscriptBox["a", "1"], "\[Equal]", SubscriptBox["a", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["a", RowBox[List["u", "+", "1"]]], "\[Equal]", RowBox[List["s", "+", "1"]]]], "\[And]", RowBox[List["s", "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", " ", "=", " ", "1"]], RowBox[List[" ", "m"]]], SubscriptBox["n", "k"]]]]], "\[And]", " ", RowBox[List["u", "\[Element]", "Integers"]], "\[And]", RowBox[List["u", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["Not", "[", RowBox[List[RowBox[List["Element", "[", RowBox[List[RowBox[List["s", "-", SubscriptBox["n", "m"]]], ",", "Integers"]], "]"]], "\[And]", RowBox[List[RowBox[List["s", "-", SubscriptBox["n", "m"]]], "\[GreaterEqual]", "0"]]]], "]"]]]]]]]]

 MathML Form

 u ( n 1 + n 2 + + n m ; n 1 , n 2 , , n m ) 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]] n m u ( - 1 ) u u ! Γ ( s + 1 ) u + 1 k = 1 m - 1 Γ ( n k + 1 ) Γ ( n m - s ) u + 2 F ~ u + 1 ( a 1 , a 2 , , a u + 1 , s - n m + 1 ; a 1 + 1 , a 2 + 1 , , a u + 1 + 1 ; 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", RowBox[List["u", "+", "2"]]], SubscriptBox[OverscriptBox["F", "~"], RowBox[List["u", "+", "1"]]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", RowBox[List["u", "+", "1"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["s", "-", SubscriptBox["n", "m"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["a", "1"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "2"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", RowBox[List["u", "+", "1"]]], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, True]], ";", TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQRegularized] /; a 1 a 2 a u + 1 s + 1 s k = 1 m n k u s - n m Condition D Multinomial Subscript n 1 Subscript n 2 Subscript n m Subscript n m u -1 u u Gamma s 1 u 1 k 1 m -1 Gamma Subscript n k 1 Gamma Subscript n m -1 s -1 HypergeometricPFQRegularized Subscript a 1 Subscript a 2 Subscript a u 1 s -1 Subscript n m 1 Subscript a 1 1 Subscript a 2 1 Subscript a u 1 1 1 Subscript a 1 Subscript a 2 Subscript a u 1 s 1 s k 1 m Subscript n k u s -1 Subscript n m [/itex]

 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