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 HypergeometricPFQ

 http://functions.wolfram.com/07.26.21.0003.01

 Input Form

 Integrate[t^(\[Alpha] - 1) HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2]}, {Subscript[b, 1], Subscript[b, 2], Subscript[b, 3]}, -t], {t, 0, Infinity}] == (Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]] Gamma[Subscript[b, 3]] Gamma[\[Alpha]] Gamma[-\[Alpha] + Subscript[a, 1]] Gamma[-\[Alpha] + Subscript[a, 2]])/(Gamma[Subscript[a, 1]] Gamma[Subscript[a, 2]] Gamma[-\[Alpha] + Subscript[b, 1]] Gamma[-\[Alpha] + Subscript[b, 2]] Gamma[-\[Alpha] + Subscript[b, 3]]) /; 0 < Re[\[Alpha]] < Min[Re[Subscript[a, 1]], Re[Subscript[a, 2]], 1/4 - (1/2) Re[Sum[Subscript[a, 1], {j, 1, 2}] - Sum[Subscript[b, k], {k, 1, 3}]]]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[SuperscriptBox["t", RowBox[List["\[Alpha]", "-", "1"]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"], ",", SubscriptBox["b", "3"]]], "}"]], ",", RowBox[List["-", "t"]]]], "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", " ", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["b", "1"], "]"]], " ", RowBox[List["Gamma", "[", SubscriptBox["b", "2"], "]"]], " ", RowBox[List["Gamma", "[", SubscriptBox["b", "3"], "]"]], RowBox[List["Gamma", "[", "\[Alpha]", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Alpha]"]], "+", SubscriptBox["a", "1"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Alpha]"]], "+", SubscriptBox["a", "2"]]], "]"]]]], "/", RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["a", "1"], "]"]], " ", RowBox[List["Gamma", "[", SubscriptBox["a", "2"], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Alpha]"]], "+", SubscriptBox["b", "1"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Alpha]"]], "+", SubscriptBox["b", "2"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Alpha]"]], "+", SubscriptBox["b", "3"]]], "]"]]]], ")"]]]]]], "/;", RowBox[List["0", "<", RowBox[List["Re", "[", "\[Alpha]", "]"]], "<", RowBox[List["Min", "[", RowBox[List[RowBox[List["Re", "[", SubscriptBox["a", "1"], "]"]], ",", RowBox[List["Re", "[", SubscriptBox["a", "2"], "]"]], ",", RowBox[List[FractionBox["1", "4"], "-", RowBox[List[FractionBox["1", "2"], RowBox[List["Re", "[", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "2"], SubscriptBox["a", "1"]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "3"], SubscriptBox["b", "k"]]]]], "]"]]]]]]]], "]"]]]]]]]]

 MathML Form

 0 t α - 1 2 F 3 ( a 1 , a 2 ; b 1 , b 2 , b 3 ; - t ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", 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]]]], 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[RowBox[List["-", "t"]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] t Γ ( b 1 ) Γ ( b 2 ) Γ ( b 3 ) Γ ( α ) Γ ( a 1 - α ) Γ ( a 2 - α ) Γ ( a 1 ) Γ ( a 2 ) Γ ( b 1 - α ) Γ ( b 2 - α ) Γ ( b 3 - α ) /; 0 < Re ( α ) < min ( Re ( a 1 ) , Re ( a 2 ) , 1 4 - 1 2 Re ( j = 1 2 a 1 - k = 1 3 b k ) ) Condition t 0 t α -1 HypergeometricPFQ Subscript a 1 Subscript a 2 Subscript b 1 Subscript b 2 Subscript b 3 -1 t Gamma Subscript b 1 Gamma Subscript b 2 Gamma Subscript b 3 Gamma α Gamma Subscript a 1 -1 α Gamma Subscript a 2 -1 α Gamma Subscript a 1 Gamma Subscript a 2 Gamma Subscript b 1 -1 α Gamma Subscript b 2 -1 α Gamma Subscript b 3 -1 α -1 0 α Subscript a 1 Subscript a 2 1 4 -1 1 2 j 1 2 Subscript a 1 -1 k 1 3 Subscript b k [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List[SuperscriptBox["t_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a_", "1"], ",", SubscriptBox["a_", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b_", "1"], ",", SubscriptBox["b_", "2"], ",", SubscriptBox["b_", "3"]]], "}"]], ",", RowBox[List["-", "t_"]]]], "]"]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["bb", "1"], "]"]], " ", RowBox[List["Gamma", "[", SubscriptBox["bb", "2"], "]"]], " ", RowBox[List["Gamma", "[", SubscriptBox["bb", "3"], "]"]], " ", RowBox[List["Gamma", "[", "\[Alpha]", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Alpha]"]], "+", SubscriptBox["aa", "1"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Alpha]"]], "+", SubscriptBox["aa", "2"]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["aa", "1"], "]"]], " ", RowBox[List["Gamma", "[", SubscriptBox["aa", "2"], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Alpha]"]], "+", SubscriptBox["bb", "1"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Alpha]"]], "+", SubscriptBox["bb", "2"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Alpha]"]], "+", SubscriptBox["bb", "3"]]], "]"]]]]], "/;", RowBox[List["0", "<", RowBox[List["Re", "[", "\[Alpha]", "]"]], "<", RowBox[List["Min", "[", RowBox[List[RowBox[List["Re", "[", SubscriptBox["aa", "1"], "]"]], ",", RowBox[List["Re", "[", SubscriptBox["aa", "2"], "]"]], ",", RowBox[List[FractionBox["1", "4"], "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Re", "[", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "2"], SubscriptBox["aa", "1"]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "3"], SubscriptBox["b", "k"]]]]], "]"]]]]]]]], "]"]]]]]]]]]]

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

 2001-10-29