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 Gamma

 http://functions.wolfram.com/06.06.06.0019.01

 Input Form

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

 Standard Form

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

 MathML Form

 Γ ( a , z ) k = 0 ( Γ ( k ) TagBox[RowBox[List["(", "k", ")"]], Derivative] ( a 0 ) k ! - z a 0 j = 0 k ( - 1 ) k - j Γ ( a 0 ) k - j + 1 log j ( z ) j ! k - j + 1 F ~ k - j + 1 ( c 1 , c 2 , , c k - j + 1 TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["c", "1"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[SubscriptBox["c", "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[SubscriptBox["c", RowBox[List["k", "-", "j", "+", "1"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, True]] ; c 1 + 1 , c 2 + 1 , , c k - j + 1 + 1 TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["c", "1"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["c", "2"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["c", RowBox[List["k", "-", "j", "+", "1"]]], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, True]] ; - z TagBox[RowBox[List["-", "z"]], HypergeometricPFQRegularized, Rule[Editable, True]] ) ) ( a - a 0 ) k /; c 1 c 2 c k + 1 a 0 k FormBox RowBox RowBox RowBox Γ ( RowBox a , z ) RowBox UnderoverscriptBox RowBox k = 0 ErrorBox RowBox RowBox ( RowBox FractionBox RowBox SuperscriptBox Γ TagBox RowBox ( k ) Derivative ( SubscriptBox a 0 ) RowBox k ! - RowBox SuperscriptBox z SubscriptBox a 0 RowBox UnderoverscriptBox RowBox j = 0 k RowBox FractionBox RowBox SuperscriptBox RowBox ( RowBox - 1 ) RowBox k - j SuperscriptBox RowBox Γ ( SubscriptBox a 0 ) RowBox k - j + 1 RowBox SuperscriptBox log j ( z ) RowBox j ! RowBox RowBox SubscriptBox RowBox k - j + 1 SubscriptBox OverscriptBox F ~ RowBox k - j + 1 RowBox ( RowBox TagBox TagBox RowBox TagBox SubscriptBox c 1 HypergeometricPFQRegularized Rule Editable , TagBox SubscriptBox c 2 HypergeometricPFQRegularized Rule Editable , TagBox HypergeometricPFQRegularized Rule Editable , TagBox SubscriptBox c RowBox k - j + 1 HypergeometricPFQRegularized Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQRegularized Rule Editable ; TagBox TagBox RowBox TagBox RowBox SubscriptBox c 1 + 1 HypergeometricPFQRegularized Rule Editable , TagBox RowBox SubscriptBox c 2 + 1 HypergeometricPFQRegularized Rule Editable , TagBox HypergeometricPFQRegularized Rule Editable , TagBox RowBox SubscriptBox c RowBox k - j + 1 + 1 HypergeometricPFQRegularized Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQRegularized Rule Editable ; TagBox RowBox - z HypergeometricPFQRegularized Rule Editable ) ) SuperscriptBox RowBox ( RowBox a - SubscriptBox a 0 ) k /; RowBox RowBox SubscriptBox c 1 SubscriptBox c 2 SubscriptBox c RowBox k + 1 SubscriptBox a 0 RowBox k TraditionalForm [/itex]

 Rule Form

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

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

 2007-05-02