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 InverseGammaRegularized

 http://functions.wolfram.com/06.12.20.0013.01

 Input Form

 D[InverseGammaRegularized[a, z], {z, n}] == KroneckerDelta[n] w + (-((Gamma[a] E^w)/w^(a - 1)))^n Sum[\[Ellipsis] Sum[KroneckerDelta[Sum[(i - 1) Subscript[j, i], {i, 2, n}], n - 1] (-1)^Sum[Subscript[j, i], {i, 2, n}] (n - 1 + Sum[Subscript[j, i], {i, 2, n}])! Product[(1/Subscript[j, i]!) ((Gamma[a + 1]/i!) E^w w^(1 - a - i))^ Subscript[j, i] Sum[(-1)^(i - k) Binomial[i, k] Pochhammer[1 - a - k, i - 1] GammaRegularized[a + k, w], {k, 0, i}]^Subscript[j, i], {i, 2, n}], {Subscript[j, n], 0, n}], {Subscript[j, 2], 0, n}] /; w == InverseGammaRegularized[a, z] && Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "n"]], "}"]]], RowBox[List["InverseGammaRegularized", "[", RowBox[List["a", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["KroneckerDelta", "[", "n", "]"]], " ", "w"]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], " ", SuperscriptBox["\[ExponentialE]", "w"]]], SuperscriptBox["w", RowBox[List["a", "-", "1"]]]]]], ")"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["j", "2"], "=", "0"]], "n"], RowBox[List["\[Ellipsis]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["j", "n"], "=", "0"]], "n"], RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "2"]], "n"], RowBox[List[RowBox[List["(", RowBox[List["i", "-", "1"]], ")"]], " ", SubscriptBox["j", "i"]]]]], ",", RowBox[List["n", "-", "1"]]]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "2"]], "n"], SubscriptBox["j", "i"]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1", "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "2"]], "n"], SubscriptBox["j", "i"]]]]], ")"]], "!"]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["i", "=", "2"]], "n"], RowBox[List[FractionBox["1", RowBox[List[SubscriptBox["j", "i"], "!"]]], SuperscriptBox[RowBox[List["(", RowBox[List[FractionBox[RowBox[List[" ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "1"]], "]"]]]], RowBox[List["i", "!"]]], SuperscriptBox["\[ExponentialE]", "w"], " ", SuperscriptBox["w", RowBox[List["1", "-", "a", "-", "i"]]]]], ")"]], SubscriptBox["j", "i"]], SuperscriptBox[RowBox[List["(", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "i"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["i", "-", "k"]]], " ", RowBox[List["Binomial", "[", RowBox[List["i", ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "a", "-", "k"]], ",", RowBox[List["i", "-", "1"]]]], "]"]], " ", RowBox[List["GammaRegularized", "[", RowBox[List[RowBox[List["a", "+", "k"]], ",", "w"]], "]"]]]]]], ")"]], SubscriptBox["j", "i"]]]]]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["w", "\[Equal]", RowBox[List["InverseGammaRegularized", "[", RowBox[List["a", ",", "z"]], "]"]]]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 n Q GammaRegularized - 1 ( a , z ) z n w δ KroneckerDelta n + ( - Γ ( a ) w w a - 1 ) n j 2 = 0 n j n = 0 n δ KroneckerDelta i = 2 n ( i - 1 ) j i , n - 1 ( - 1 ) i = 2 n j i ( n + i = 2 n j i - 1 ) ! i = 2 n 1 j i ! ( Γ ( a + 1 ) w w - a - i + 1 i ! ) j i ( k = 0 i ( - 1 ) i - k ( i k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["i", Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( - a - k + 1 ) i - 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "-", "k", "+", "1"]], ")"]], RowBox[List["i", "-", "1"]]], Pochhammer] Q GammaRegularized ( a + k , w ) ) j i /; w Q GammaRegularized - 1 ( a , z ) n Condition z n InverseGammaRegularized a z w KroneckerDelta n -1 Gamma a w w a -1 -1 n Subscript j 2 0 n Subscript j n 0 n KroneckerDelta i 2 n i -1 Subscript j i n -1 -1 i 2 n Subscript j i n i 2 n Subscript j i -1 i 2 n 1 Subscript j i -1 Gamma a 1 w w -1 a -1 i 1 i -1 Subscript j i k 0 i -1 i -1 k Binomial i k Pochhammer -1 a -1 k 1 i -1 GammaRegularized a k w Subscript j i w InverseGammaRegularized a z n [/itex]

 Rule Form

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

 2007-05-02