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 GammaRegularized

 http://functions.wolfram.com/06.08.06.0013.01

 Input Form

 GammaRegularized[a, z] == Sum[(Derivative[k - s][Gamma][Subscript[a, 0]] - z^Subscript[a, 0] Sum[(-1)^(k - s - i) Binomial[k - s, i] (k - s - i)! Gamma[Subscript[a, 0]]^(k - s - i + 1) Log[z]^i HypergeometricPFQRegularized[{Subscript[c, 1], Subscript[c, 2], \[Ellipsis], Subscript[c, k - s - i + 1]}, {1 + Subscript[c, 1], 1 + Subscript[c, 2], \[Ellipsis], 1 + Subscript[c, k - s - i + 1]}, -z], {i, 0, k - s}]) Sum[(((-1)^j (s + 1) Gamma[Subscript[a, 0]]^(-j - 1))/ ((j + 1)! (k - s)! (s - j)!)) Derivative[s][Function[u, Gamma[u]^j]][ Subscript[a, 0]] (a - Subscript[a, 0])^k, {j, 0, s}], {k, 0, Infinity}, {s, 0, k}] /; 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["GammaRegularized", "[", RowBox[List["a", ",", "z"]], "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], "k"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["Gamma", TagBox[RowBox[List["(", RowBox[List["k", "-", "s"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", SubscriptBox["a", "0"], "]"]], "-", RowBox[List[SuperscriptBox["z", SubscriptBox["a", "0"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["k", "-", "s"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "s", "-", "i"]]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["k", "-", "s"]], ",", "i"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "s", "-", "i"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", SubscriptBox["a", "0"], "]"]], RowBox[List["k", "-", "s", "-", "i", "+", "1"]]], " ", SuperscriptBox[RowBox[List["Log", "[", "z", "]"]], "i"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["c", "1"], ",", SubscriptBox["c", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["c", RowBox[List["k", "-", "s", "-", "i", "+", "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", "-", "s", "-", "i", "+", "1"]]]]]]], "}"]], ",", RowBox[List["-", "z"]]]], "]"]]]]]]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "s"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["(", RowBox[List["s", "+", "1"]], ")"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", SubscriptBox["a", "0"], "]"]], RowBox[List[RowBox[List["-", "j"]], "-", "1"]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["j", "+", "1"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "s"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["s", "-", "j"]], ")"]], "!"]]]]], RowBox[List[RowBox[List[RowBox[List["Derivative", "[", "s", "]"]], "[", RowBox[List["Function", "[", RowBox[List["u", ",", " ", SuperscriptBox[RowBox[List["Gamma", "[", "u", "]"]], "j"]]], "]"]], "]"]], "[", SubscriptBox["a", "0"], "]"]], 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

 Q GammaRegularized ( a , z ) k = 0 s = 0 k ( Γ ( k - s ) TagBox[RowBox[List["(", RowBox[List["k", "-", "s"]], ")"]], Derivative] ( a 0 ) - z a 0 i = 0 k - s ( - 1 ) k - i - s ( k - s i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["k", "-", "s"]], Identity, Rule[Editable, True]]], List[TagBox["i", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( k - i - s ) ! Γ ( a 0 ) k - i - s + 1 log i ( z ) 4 F ~ 4 ( c 1 , c 2 , , c k - i - s + 1 ; c 1 + 1 , c 2 + 1 , , c k - i - s + 1 + 1 ; - z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["4", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["4", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[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", "-", "i", "-", "s", "+", "1"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", 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", "-", "i", "-", "s", "+", "1"]]], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[RowBox[List["-", "z"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] ) j = 0 s ( - 1 ) j ( s + 1 ) Γ ( a 0 ) - j - 1 ( j + 1 ) ! ( k - s ) ! ( s - j ) ! Function [ u , Γ ( u ) j ] ( s ) TagBox[RowBox[List["(", "s", ")"]], Derivative] ( a 0 ) ( a - a 0 ) k /; c 1 c 2 c k + 1 a 0 k Condition GammaRegularized a z s 0 k k 0 D Gamma Subscript a 0 Subscript a 0 k -1 s -1 z Subscript a 0 i 0 k -1 s -1 k -1 i -1 s Binomial k -1 s i k -1 i -1 s Gamma Subscript a 0 k -1 i -1 s 1 z i HypergeometricPFQRegularized Subscript c 1 Subscript c 2 Subscript c k -1 i -1 s 1 Subscript c 1 1 Subscript c 2 1 Subscript c k -1 i -1 s 1 1 -1 z j 0 s -1 j s 1 Gamma Subscript a 0 -1 j -1 j 1 k -1 s s -1 j -1 D Function u Gamma u j Subscript a 0 Subscript a 0 s a -1 Subscript a 0 k Subscript c 1 Subscript c 2 Subscript c k 1 Subscript a 0 k [/itex]

 Rule Form

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