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 HypergeometricPFQ

 http://functions.wolfram.com/07.27.22.0001.01

 Input Form

 LaplaceTransform[HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3]}, {Subscript[b, 1], Subscript[b, 2]}, -t], t, z] == (Pi z^(-1 + Subscript[a, 1]) Csc[Pi Subscript[a, 1]] Gamma[-Subscript[a, 1] + Subscript[a, 2]] Gamma[-Subscript[a, 1] + Subscript[a, 3]] Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]] HypergeometricPFQ[ {1 + Subscript[a, 1] - Subscript[b, 1], 1 + Subscript[a, 1] - Subscript[b, 2]}, {1 + Subscript[a, 1] - Subscript[a, 2], 1 + Subscript[a, 1] - Subscript[a, 3]}, z])/ (Gamma[Subscript[a, 1]] Gamma[Subscript[a, 2]] Gamma[Subscript[a, 3]] Gamma[-Subscript[a, 1] + Subscript[b, 1]] Gamma[-Subscript[a, 1] + Subscript[b, 2]]) + (Pi z^(-1 + Subscript[a, 2]) Csc[Pi Subscript[a, 2]] Gamma[Subscript[a, 1] - Subscript[a, 2]] Gamma[-Subscript[a, 2] + Subscript[a, 3]] Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]] HypergeometricPFQ[ {1 + Subscript[a, 2] - Subscript[b, 1], 1 + Subscript[a, 2] - Subscript[b, 2]}, {1 - Subscript[a, 1] + Subscript[a, 2], 1 + Subscript[a, 2] - Subscript[a, 3]}, z])/ (Gamma[Subscript[a, 1]] Gamma[Subscript[a, 2]] Gamma[Subscript[a, 3]] Gamma[-Subscript[a, 2] + Subscript[b, 1]] Gamma[-Subscript[a, 2] + Subscript[b, 2]]) + (Pi z^(-1 + Subscript[a, 3]) Csc[Pi Subscript[a, 3]] Gamma[Subscript[a, 1] - Subscript[a, 3]] Gamma[Subscript[a, 2] - Subscript[a, 3]] Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]] HypergeometricPFQ[ {1 + Subscript[a, 3] - Subscript[b, 1], 1 + Subscript[a, 3] - Subscript[b, 2]}, {1 - Subscript[a, 1] + Subscript[a, 3], 1 - Subscript[a, 2] + Subscript[a, 3]}, z])/ (Gamma[Subscript[a, 1]] Gamma[Subscript[a, 2]] Gamma[Subscript[a, 3]] Gamma[-Subscript[a, 3] + Subscript[b, 1]] Gamma[-Subscript[a, 3] + Subscript[b, 2]]) + (HypergeometricPFQ[{1, 2 - Subscript[b, 1], 2 - Subscript[b, 2]}, {2 - Subscript[a, 1], 2 - Subscript[a, 2], 2 - Subscript[a, 3]}, z] (-1 + Subscript[b, 1]) (-1 + Subscript[b, 2]))/ ((-1 + Subscript[a, 1]) (-1 + Subscript[a, 2]) (-1 + Subscript[a, 3])) /; Re[z] > 0

 Standard Form

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 MathML Form

 t [ 3 F 2 ( a 1 , a 2 , a 3 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]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]] ; b 1 , b 2 TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["b", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "2"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]] ; - t TagBox[RowBox[List["-", "t"]], HypergeometricPFQ, Rule[Editable, True]] ) ] ( z ) π csc ( π a 1 ) Γ ( a 2 - a 1 ) Γ ( a 3 - a 1 ) Γ ( b 1 ) Γ ( b 2 ) Γ ( a 1 ) Γ ( a 2 ) Γ ( a 3 ) Γ ( b 1 - a 1 ) Γ ( b 2 - a 1 ) z a 1 - 1 2 F 2 ( a 1 - b 1 + 1 , a 1 - b 2 + 1 ; a 1 - a 2 + 1 , a 1 - a 3 + 1 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["a", "1"], "-", SubscriptBox["b", "1"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "1"], "-", SubscriptBox["b", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["a", "1"], "-", SubscriptBox["a", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "1"], "-", SubscriptBox["a", "3"], "+", "1"]], 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] + π csc ( π a 2 ) Γ ( a 1 - a 2 ) Γ ( a 3 - a 2 ) Γ ( b 1 ) Γ ( b 2 ) Γ ( a 1 ) Γ ( a 2 ) Γ ( a 3 ) Γ ( b 1 - a 2 ) Γ ( b 2 - a 2 ) z a 2 - 1 2 F 2 ( a 2 - b 1 + 1 , a 2 - b 2 + 1 ; - a 1 + a 2 + 1 , a 2 - a 3 + 1 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["a", "2"], "-", SubscriptBox["b", "1"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "2"], "-", SubscriptBox["b", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[RowBox[List["-", SubscriptBox["a", "1"]]], "+", SubscriptBox["a", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "2"], "-", SubscriptBox["a", "3"], "+", "1"]], 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] + π csc ( π a 3 ) Γ ( a 1 - a 3 ) Γ ( a 2 - a 3 ) Γ ( b 1 ) Γ ( b 2 ) Γ ( a 1 ) Γ ( a 2 ) Γ ( a 3 ) Γ ( b 1 - a 3 ) Γ ( b 2 - a 3 ) z a 3 - 1 2 F 2 ( a 3 - b 1 + 1 , a 3 - b 2 + 1 ; - a 1 + a 3 + 1 , - a 2 + a 3 + 1 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["a", "3"], "-", SubscriptBox["b", "1"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "3"], "-", SubscriptBox["b", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[RowBox[List["-", SubscriptBox["a", "1"]]], "+", SubscriptBox["a", "3"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", SubscriptBox["a", "2"]]], "+", SubscriptBox["a", "3"], "+", "1"]], 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] + ( b 1 - 1 ) ( b 2 - 1 ) ( a 1 - 1 ) ( a 2 - 1 ) ( a 3 - 1 ) 3 F 3 ( 1 , 2 - b 1 , 2 - b 2 ; 2 - a 1 , 2 - a 2 , 2 - a 3 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["3", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["2", "-", SubscriptBox["b", "1"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["2", "-", SubscriptBox["b", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["2", "-", SubscriptBox["a", "1"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["2", "-", SubscriptBox["a", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["2", "-", SubscriptBox["a", "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] /; Re ( z ) > 0 t [ 3 F 2 ( a 1 , a 2 , a 3 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]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]] ; b 1 , b 2 TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["b", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "2"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]] ; - t TagBox[RowBox[List["-", "t"]], HypergeometricPFQ, Rule[Editable, True]] ) ] ( z ) π csc ( π a 1 ) Γ ( a 2 - a 1 ) Γ ( a 3 - a 1 ) Γ ( b 1 ) Γ ( b 2 ) Γ ( a 1 ) Γ ( a 2 ) Γ ( a 3 ) Γ ( b 1 - a 1 ) Γ ( b 2 - a 1 ) z a 1 - 1 2 F 2 ( a 1 - b 1 + 1 , a 1 - b 2 + 1 ; a 1 - a 2 + 1 , a 1 - a 3 + 1 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["a", "1"], "-", SubscriptBox["b", "1"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "1"], "-", SubscriptBox["b", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["a", "1"], "-", SubscriptBox["a", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "1"], "-", SubscriptBox["a", "3"], "+", "1"]], 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] + π csc ( π a 2 ) Γ ( a 1 - a 2 ) Γ ( a 3 - a 2 ) Γ ( b 1 ) Γ ( b 2 ) Γ ( a 1 ) Γ ( a 2 ) Γ ( a 3 ) Γ ( b 1 - a 2 ) Γ ( b 2 - a 2 ) z a 2 - 1 2 F 2 ( a 2 - b 1 + 1 , a 2 - b 2 + 1 ; - a 1 + a 2 + 1 , a 2 - a 3 + 1 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["a", "2"], "-", SubscriptBox["b", "1"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "2"], "-", SubscriptBox["b", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[RowBox[List["-", SubscriptBox["a", "1"]]], "+", SubscriptBox["a", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "2"], "-", SubscriptBox["a", "3"], "+", "1"]], 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] + π csc ( π a 3 ) Γ ( a 1 - a 3 ) Γ ( a 2 - a 3 ) Γ ( b 1 ) Γ ( b 2 ) Γ ( a 1 ) Γ ( a 2 ) Γ ( a 3 ) Γ ( b 1 - a 3 ) Γ ( b 2 - a 3 ) z a 3 - 1 2 F 2 ( a 3 - b 1 + 1 , a 3 - b 2 + 1 ; - a 1 + a 3 + 1 , - a 2 + a 3 + 1 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["a", "3"], "-", SubscriptBox["b", "1"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "3"], "-", SubscriptBox["b", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[RowBox[List["-", SubscriptBox["a", "1"]]], "+", SubscriptBox["a", "3"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", SubscriptBox["a", "2"]]], "+", SubscriptBox["a", "3"], "+", "1"]], 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] + ( b 1 - 1 ) ( b 2 - 1 ) ( a 1 - 1 ) ( a 2 - 1 ) ( a 3 - 1 ) 3 F 3 ( 1 , 2 - b 1 , 2 - b 2 ; 2 - a 1 , 2 - a 2 , 2 - a 3 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["3", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["2", "-", SubscriptBox["b", "1"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["2", "-", SubscriptBox["b", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["2", "-", SubscriptBox["a", "1"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["2", "-", SubscriptBox["a", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["2", "-", SubscriptBox["a", "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] /; Re ( z ) > 0 [/itex]

 Rule Form

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

 2001-10-29