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 InverseGammaRegularized

 http://functions.wolfram.com/06.13.20.0007.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List[SubscriptBox["z", "2"], ",", "n"]], "}"]]], RowBox[List["InverseGammaRegularized", "[", RowBox[List["a", ",", SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"]]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["KroneckerDelta", "[", "n", "]"]], " ", "w"]], "+", RowBox[List[SuperscriptBox[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"]]]]], ")"]], SubscriptBox["j", "i"]], SuperscriptBox[RowBox[List["(", " ", RowBox[List[RowBox[List[RowBox[List["GammaRegularized", "[", RowBox[List["a", ",", SubscriptBox["z", "1"]]], "]"]], " ", RowBox[List["KroneckerDelta", "[", "i", "]"]]]], "+", RowBox[List["a", " ", SuperscriptBox["w", RowBox[List["-", "i"]]], " ", 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", ",", SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"]]], "]"]]]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 n Q GammaRegularized - 1 ( a , z 1 , z 2 ) z 2 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 1 - a i ! ) j i ( a w - 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]] ( 1 - a - k ) i - 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "a", "-", "k"]], ")"]], RowBox[List["i", "-", "1"]]], Pochhammer] Q GammaRegularized ( a + k , w ) + Q GammaRegularized ( a , z 1 ) δ KroneckerDelta i ) j i /; w Q GammaRegularized - 1 ( a , z 1 , z 2 ) n FormBox RowBox RowBox FractionBox RowBox SuperscriptBox n RowBox SuperscriptBox InterpretationBox Q GammaRegularized Rule Editable Rule Selectable RowBox - 1 ( RowBox a , SubscriptBox z 1 , SubscriptBox z 2 ) RowBox SubsuperscriptBox z 2 n RowBox RowBox w SubscriptBox InterpretationBox δ KroneckerDelta Rule Editable Rule Selectable n + RowBox SuperscriptBox ErrorBox RowBox ` [ FractionBox RowBox RowBox Γ ( a ) SuperscriptBox w SuperscriptBox w RowBox a - 1 ] n RowBox UnderoverscriptBox RowBox SubscriptBox j 2 = 0 n RowBox RowBox UnderoverscriptBox RowBox SubscriptBox j n = 0 n RowBox SubscriptBox InterpretationBox δ KroneckerDelta Rule Editable Rule Selectable RowBox RowBox UnderoverscriptBox RowBox i = 2 n RowBox RowBox ( RowBox i - 1 ) SubscriptBox j i , RowBox n - 1 SuperscriptBox RowBox ( RowBox - 1 ) RowBox UnderoverscriptBox RowBox i = 2 n SubscriptBox j i RowBox RowBox ( RowBox n + RowBox UnderoverscriptBox RowBox i = 2 n SubscriptBox j i - 1 ) ! RowBox UnderoverscriptBox RowBox i = 2 n RowBox FractionBox 1 RowBox SubscriptBox j i ! SuperscriptBox RowBox ( FractionBox RowBox RowBox Γ ( RowBox a + 1 ) SuperscriptBox w SuperscriptBox w RowBox 1 - a RowBox i ! ) SubscriptBox j i SuperscriptBox RowBox ( RowBox RowBox a SuperscriptBox w RowBox - i RowBox UnderoverscriptBox RowBox k = 0 i RowBox SuperscriptBox RowBox ( RowBox - 1 ) RowBox i - k TagBox RowBox ( GridBox TagBox i Rule Editable TagBox k Rule Editable ) InterpretTemplate Function Binomial Slot 1 Slot 2 Rule Editable TagBox SubscriptBox RowBox ( RowBox 1 - a - k ) RowBox i - 1 Pochhammer RowBox InterpretationBox Q GammaRegularized Rule Editable Rule Selectable ( RowBox RowBox a + k , w ) + RowBox RowBox InterpretationBox Q GammaRegularized Rule Editable Rule Selectable ( RowBox a , SubscriptBox z 1 ) SubscriptBox InterpretationBox δ KroneckerDelta Rule Editable Rule Selectable i ) SubscriptBox j i /; RowBox RowBox w RowBox SuperscriptBox InterpretationBox Q GammaRegularized Rule Editable Rule Selectable RowBox - 1 ( RowBox a , SubscriptBox z 1 , SubscriptBox z 2 ) RowBox n TraditionalForm [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["z_", "2"], ",", "n_"]], "}"]]]]], RowBox[List["InverseGammaRegularized", "[", RowBox[List["a_", ",", SubscriptBox["z_", "1"], ",", SubscriptBox["z_", "2"]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["KroneckerDelta", "[", "n", "]"]], " ", "w"]], "+", RowBox[List[SuperscriptBox[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"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["a", "+", "1"]], "]"]], " ", SuperscriptBox["\[ExponentialE]", "w"], " ", SuperscriptBox["w", RowBox[List["1", "-", "a"]]]]], RowBox[List["i", "!"]]], ")"]], SubscriptBox["j", "i"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["GammaRegularized", "[", RowBox[List["a", ",", SubscriptBox["zz", "1"]]], "]"]], " ", RowBox[List["KroneckerDelta", "[", "i", "]"]]]], "+", RowBox[List["a", " ", SuperscriptBox["w", RowBox[List["-", "i"]]], " ", 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[SubscriptBox["j", "i"], "!"]]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["w", "\[Equal]", RowBox[List["InverseGammaRegularized", "[", RowBox[List["a", ",", SubscriptBox["zz", "1"], ",", SubscriptBox["z", "2"]]], "]"]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02