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 GammaRegularized

 http://functions.wolfram.com/06.08.20.0006.02

 Input Form

 D[GammaRegularized[a, z], {a, n}] == n! Sum[(Derivative[n - k][Gamma][a] - z^a Sum[(-1)^(n - k - i) Binomial[n - k, i] (n - k - i)! Gamma[a]^(n - k - i + 1) Log[z]^i HypergeometricPFQRegularized[ {Subscript[a, 1], Subscript[a, 2], \[Ellipsis], Subscript[a, n - k - i + 1]}, {1 + Subscript[a, 1], 1 + Subscript[a, 2], \[Ellipsis], 1 + Subscript[a, n - k - i + 1]}, -z], {i, 0, n - k}]) Sum[((-1)^j (k + 1) Gamma[a]^(-j - 1) D[Gamma[a]^j, {a, k}])/ ((j + 1)! (n - k)! (k - j)!), {j, 0, k}], {k, 0, n}] /; Subscript[a, 1] == Subscript[a, 2] == \[Ellipsis] == Subscript[a, n + 1] == a && Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["a", ",", "n"]], "}"]]], RowBox[List["GammaRegularized", "[", RowBox[List["a", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List["n", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["Gamma", TagBox[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", "a", "]"]], "-", RowBox[List[SuperscriptBox["z", "a"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["n", "-", "k"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "k", "-", "i"]]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "k"]], ",", "i"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k", "-", "i"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", "a", "]"]], RowBox[List["n", "-", "k", "-", "i", "+", "1"]]], " ", SuperscriptBox[RowBox[List["Log", "[", "z", "]"]], "i"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["n", "-", "k", "-", "i", "+", "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["n", "-", "k", "-", "i", "+", "1"]]]]]]], "}"]], ",", RowBox[List["-", "z"]]]], "]"]]]]]]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", "a", "]"]], RowBox[List[RowBox[List["-", "j"]], "-", "1"]]], " ", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["a", ",", "k"]], "}"]]], SuperscriptBox[RowBox[List["Gamma", "[", "a", "]"]], "j"]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["j", "+", "1"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", "1"], "\[Equal]", SubscriptBox["a", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["a", RowBox[List["n", "+", "1"]]], "\[Equal]", "a"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 n Q GammaRegularized ( a , z ) a n n ! k = 0 n ( Γ ( n ) TagBox[RowBox[List["(", "n", ")"]], Derivative] ( a ) - z a i = 0 n - k ( - 1 ) n - i - k ( n - k i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", "k"]], Identity, Rule[Editable, True]]], List[TagBox["i", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( n - i - k ) ! Γ ( a ) n - i - k + 1 log i ( z ) n - k - i + 1 F ~ n - k - i + 1 ( a 1 , a 2 , , a n - k - i + 1 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["n", "-", "k", "-", "i", "+", "1"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, True]] ; a 1 + 1 , a 2 + 1 , , a n - k - i + 1 + 1 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["n", "-", "k", "-", "i", "+", "1"]]], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, True]] ; - z TagBox[RowBox[List["-", "z"]], HypergeometricPFQRegularized, Rule[Editable, True]] ) ) j = 0 k ( - 1 ) j ( k + 1 ) Γ ( a ) - j - 1 ( j + 1 ) ! ( n - k ) ! ( k - j ) ! k Γ ( a ) j a k /; a 1 a 2 a n + 1 a n FormBox RowBox RowBox FractionBox RowBox SuperscriptBox n RowBox InterpretationBox Q GammaRegularized Rule Editable Rule Selectable ( RowBox a , z ) RowBox SuperscriptBox a n RowBox RowBox n ! RowBox UnderoverscriptBox RowBox k = 0 n ErrorBox RowBox RowBox ( RowBox RowBox SuperscriptBox Γ TagBox RowBox ( n ) Derivative ( a ) - RowBox SuperscriptBox z a RowBox UnderoverscriptBox RowBox i = 0 RowBox n - k RowBox SuperscriptBox RowBox ( RowBox - 1 ) RowBox n - i - k TagBox RowBox ( GridBox TagBox RowBox n - k Rule Editable TagBox i Rule Editable ) InterpretTemplate Function Binomial Slot 1 Slot 2 Rule Editable RowBox RowBox ( RowBox n - i - k ) ! SuperscriptBox RowBox Γ ( a ) RowBox n - i - k + 1 RowBox SuperscriptBox log i ( z ) RowBox RowBox SubscriptBox RowBox n - k - i + 1 SubscriptBox OverscriptBox F ~ RowBox n - k - i + 1 RowBox ( RowBox TagBox TagBox RowBox TagBox SubscriptBox a 1 HypergeometricPFQRegularized Rule Editable , TagBox SubscriptBox a 2 HypergeometricPFQRegularized Rule Editable , TagBox HypergeometricPFQRegularized Rule Editable , TagBox SubscriptBox a RowBox n - k - i + 1 HypergeometricPFQRegularized Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQRegularized Rule Editable ; TagBox TagBox RowBox TagBox RowBox SubscriptBox a 1 + 1 HypergeometricPFQRegularized Rule Editable , TagBox RowBox SubscriptBox a 2 + 1 HypergeometricPFQRegularized Rule Editable , TagBox HypergeometricPFQRegularized Rule Editable , TagBox RowBox SubscriptBox a RowBox n - k - i + 1 + 1 HypergeometricPFQRegularized Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQRegularized Rule Editable ; TagBox RowBox - z HypergeometricPFQRegularized Rule Editable ) ) RowBox UnderoverscriptBox RowBox j = 0 k RowBox FractionBox RowBox SuperscriptBox RowBox ( RowBox - 1 ) j RowBox ( RowBox k + 1 ) SuperscriptBox RowBox Γ ( a ) RowBox RowBox - j - 1 RowBox RowBox RowBox ( RowBox j + 1 ) ! RowBox RowBox ( RowBox n - k ) ! RowBox RowBox ( RowBox k - j ) ! FractionBox RowBox SuperscriptBox k SuperscriptBox RowBox Γ ( a ) j RowBox SuperscriptBox a k /; RowBox RowBox SubscriptBox a 1 SubscriptBox a 2 SubscriptBox a RowBox n + 1 a RowBox n TraditionalForm [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["a_", ",", "n_"]], "}"]]]]], RowBox[List["GammaRegularized", "[", RowBox[List["a_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["n", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["Gamma", TagBox[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", "a", "]"]], "-", RowBox[List[SuperscriptBox["z", "a"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["n", "-", "k"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "k", "-", "i"]]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "k"]], ",", "i"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k", "-", "i"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", "a", "]"]], RowBox[List["n", "-", "k", "-", "i", "+", "1"]]], " ", SuperscriptBox[RowBox[List["Log", "[", "z", "]"]], "i"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["aa", "1"], ",", SubscriptBox["aa", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["n", "-", "k", "-", "i", "+", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["aa", "1"]]], ",", RowBox[List["1", "+", SubscriptBox["aa", "2"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["a", RowBox[List["n", "-", "k", "-", "i", "+", "1"]]]]]]], "}"]], ",", RowBox[List["-", "z"]]]], "]"]]]]]]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", "a", "]"]], RowBox[List[RowBox[List["-", "j"]], "-", "1"]]], " ", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["a", ",", "k"]], "}"]]]]], SuperscriptBox[RowBox[List["Gamma", "[", "a", "]"]], "j"]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["j", "+", "1"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["aa", "1"], "\[Equal]", SubscriptBox["aa", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["a", RowBox[List["n", "+", "1"]]], "\[Equal]", "a"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2001-10-29