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 HypergeometricPFQ

 http://functions.wolfram.com/07.28.07.0005.01

 Input Form

 HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3], Subscript[a, 4]}, {Subscript[b, 1], Subscript[b, 2], Subscript[b, 3]}, z] == Product[(Gamma[Subscript[b, k]]/(Gamma[Subscript[a, k]] Gamma[Subscript[b, k] - Subscript[a, k]])) Integrate[Product[(Subscript[t, k]^(Subscript[a, k] - 1) (1 - Subscript[t, k])^(Subscript[b, k] - Subscript[a, k] - 1))/ (1 - z Product[Subscript[t, k], {k, 1, 3}])^Subscript[a, 4], {k, 1, 3}], {Subscript[t, 3], 0, 1}, {Subscript[t, 2], 0, 1}, {Subscript[t, 1], 0, 1}], {k, 1, 3}] /; Re[Subscript[b, k]] > Re[Subscript[a, k]] > 0 && 1 <= k <= 3 && Abs[Arg[1 - z]] < Pi

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", SubscriptBox["a", "3"], ",", SubscriptBox["a", "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"], ",", SubscriptBox["b", "3"]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "3"], RowBox[List[FractionBox[RowBox[List[" ", RowBox[List["Gamma", "[", SubscriptBox["b", "k"], "]"]]]], RowBox[List[" ", RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["a", "k"], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "k"], "-", SubscriptBox["a", "k"]]], "]"]]]]]]], RowBox[List[SubsuperscriptBox["\[Integral]", "0", "1"], RowBox[List[SubsuperscriptBox["\[Integral]", "0", "1"], RowBox[List[SubsuperscriptBox["\[Integral]", "0", "1"], RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "3"], RowBox[List[SubsuperscriptBox["t", "k", RowBox[List[SubscriptBox["a", "k"], "-", "1"]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["t", "k"]]], ")"]], RowBox[List[SubscriptBox["b", "k"], "-", SubscriptBox["a", "k"], "-", "1"]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["z", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "3"], SubscriptBox["t", "k"]]]]]]], ")"]], RowBox[List["-", SubscriptBox["a", "4"]]]]]]]], ")"]], RowBox[List["\[DifferentialD]", SubscriptBox["t", "1"]]], RowBox[List["\[DifferentialD]", SubscriptBox["t", "2"]]], RowBox[List["\[DifferentialD]", SubscriptBox["t", "3"]]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", SubscriptBox["b", "k"], "]"]], ">", RowBox[List["Re", "[", SubscriptBox["a", "k"], "]"]], ">", "0"]], "\[And]", RowBox[List["1", "\[LessEqual]", "k", "\[LessEqual]", "3"]], "\[And]", RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", RowBox[List["1", "-", "z"]], "]"]], "]"]], "<", "\[Pi]"]]]]]]]]

 MathML Form

 4 F 3 ( a 1 , a 2 , a 3 , a 4 ; b 1 , b 2 , b 3 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["4", TraditionalForm]], SubscriptBox["F", FormBox["3", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "3"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "4"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["b", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "3"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox["z", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ( k = 1 3 Γ ( b k ) Γ ( a k ) Γ ( b k - a k ) ) 0 1 0 1 0 1 k = 1 3 t k a k - 1 ( 1 - t k ) - a k + b k - 1 ( 1 - z k = 1 3 t k ) - a 4 t 1 t 2 t 3 /; Re ( b k ) > Re ( a k ) > 0 1 k 3 "\[LeftBracketingBar]" arg ( 1 - z ) "\[RightBracketingBar]" < π Condition HypergeometricPFQ Subscript a 1 Subscript a 2 Subscript a 3 Subscript a 4 Subscript b 1 Subscript b 2 Subscript b 3 z k 1 3 Gamma Subscript b k Gamma Subscript a k Gamma Subscript b k -1 Subscript a k -1 Subscript t 1 0 1 Subscript t 2 0 1 Subscript t 3 0 1 k 1 3 Subscript t k Subscript a k -1 1 -1 Subscript t k -1 Subscript a k Subscript b k -1 1 -1 z k 1 3 Subscript t k -1 Subscript a 4 Subscript b k Subscript a k 0 1 k 3 1 -1 z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a_", "1"], ",", SubscriptBox["a_", "2"], ",", SubscriptBox["a_", "3"], ",", SubscriptBox["a_", "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b_", "1"], ",", SubscriptBox["b_", "2"], ",", SubscriptBox["b_", "3"]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "3"], FractionBox[RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["b", "k"], "]"]], " ", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "1"], RowBox[List[SubsuperscriptBox["\[Integral]", "0", "1"], RowBox[List[SubsuperscriptBox["\[Integral]", "0", "1"], RowBox[List[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "3"], RowBox[List[SubsuperscriptBox["t", "k", RowBox[List[SubscriptBox["a", "k"], "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["t", "k"]]], ")"]], RowBox[List[SubscriptBox["b", "k"], "-", SubscriptBox["a", "k"], "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["z", " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "3"], SubscriptBox["t", "k"]]]]]]], ")"]], RowBox[List["-", SubscriptBox["aa", "4"]]]]]]]], RowBox[List["\[DifferentialD]", SubscriptBox["t", "1"]]], RowBox[List["\[DifferentialD]", SubscriptBox["t", "2"]]], RowBox[List["\[DifferentialD]", SubscriptBox["t", "3"]]]]]]]]]]]]], RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["a", "k"], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "k"], "-", SubscriptBox["a", "k"]]], "]"]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", SubscriptBox["b", "k"], "]"]], ">", RowBox[List["Re", "[", SubscriptBox["a", "k"], "]"]], ">", "0"]], "&&", RowBox[List["1", "\[LessEqual]", "k", "\[LessEqual]", "3"]], "&&", RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", RowBox[List["1", "-", "z"]], "]"]], "]"]], "<", "\[Pi]"]]]]]]]]]]

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

 2001-10-29