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 HypergeometricPFQ

 http://functions.wolfram.com/07.27.21.0006.01

 Input Form

 Integrate[(t^(\[Alpha] - 1) HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3]}, {Subscript[b, 1], Subscript[b, 2]}, -t])/E^(c t), {t, 0, Infinity}] == (Gamma[\[Alpha]] Gamma[-\[Alpha] + Subscript[a, 1]] Gamma[-\[Alpha] + Subscript[a, 2]] Gamma[-\[Alpha] + Subscript[a, 3]] Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]] HypergeometricPFQ[{\[Alpha], 1 + \[Alpha] - Subscript[b, 1], 1 + \[Alpha] - Subscript[b, 2]}, {1 + \[Alpha] - Subscript[a, 1], 1 + \[Alpha] - Subscript[a, 2], 1 + \[Alpha] - Subscript[a, 3]}, c])/ (Gamma[Subscript[a, 1]] Gamma[Subscript[a, 2]] Gamma[Subscript[a, 3]] Gamma[-\[Alpha] + Subscript[b, 1]] Gamma[-\[Alpha] + Subscript[b, 2]]) + (c^(-\[Alpha] + Subscript[a, 1]) Gamma[\[Alpha] - 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[{Subscript[a, 1], 1 + Subscript[a, 1] - Subscript[b, 1], 1 + Subscript[a, 1] - Subscript[b, 2]}, {1 - \[Alpha] + Subscript[a, 1], 1 + Subscript[a, 1] - Subscript[a, 2], 1 + Subscript[a, 1] - Subscript[a, 3]}, c])/(Gamma[Subscript[a, 2]] Gamma[Subscript[a, 3]] Gamma[-Subscript[a, 1] + Subscript[b, 1]] Gamma[-Subscript[a, 1] + Subscript[b, 2]]) + (c^(-\[Alpha] + Subscript[a, 2]) Gamma[\[Alpha] - 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[{Subscript[a, 2], 1 + Subscript[a, 2] - Subscript[b, 1], 1 + Subscript[a, 2] - Subscript[b, 2]}, {1 - \[Alpha] + Subscript[a, 2], 1 - Subscript[a, 1] + Subscript[a, 2], 1 + Subscript[a, 2] - Subscript[a, 3]}, c])/(Gamma[Subscript[a, 1]] Gamma[Subscript[a, 3]] Gamma[-Subscript[a, 2] + Subscript[b, 1]] Gamma[-Subscript[a, 2] + Subscript[b, 2]]) + (c^(-\[Alpha] + Subscript[a, 3]) Gamma[\[Alpha] - 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[{Subscript[a, 3], 1 + Subscript[a, 3] - Subscript[b, 1], 1 + Subscript[a, 3] - Subscript[b, 2]}, {1 - \[Alpha] + Subscript[a, 3], 1 - Subscript[a, 1] + Subscript[a, 3], 1 - Subscript[a, 2] + Subscript[a, 3]}, c])/(Gamma[Subscript[a, 1]] Gamma[Subscript[a, 2]] Gamma[-Subscript[a, 3] + Subscript[b, 1]] Gamma[-Subscript[a, 3] + Subscript[b, 2]]) /; Re[\[Alpha]] > 0 && Re[c] > 0

 Standard Form

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 MathML Form

 0 t α - 1 - c t 3 F 2 ( a 1 , a 2 , a 3 ; b 1 , b 2 ; - t ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["2", 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]]]], 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]]]], 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 ( α ) Γ ( a 1 - α ) Γ ( a 2 - α ) Γ ( a 3 - α ) Γ ( b 1 ) Γ ( b 2 ) Γ ( a 1 ) Γ ( a 2 ) Γ ( a 3 ) Γ ( b 1 - α ) Γ ( b 2 - α ) 3 F 3 ( α , α - b 1 + 1 , α - b 2 + 1 ; α - a 1 + 1 , α - a 2 + 1 , α - a 3 + 1 ; c ) 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["\[Alpha]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Alpha]", "-", SubscriptBox["b", "1"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Alpha]", "-", SubscriptBox["b", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["\[Alpha]", "-", SubscriptBox["a", "1"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Alpha]", "-", SubscriptBox["a", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Alpha]", "-", SubscriptBox["a", "3"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox["c", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] + Γ ( α - a 1 ) Γ ( a 2 - a 1 ) Γ ( a 3 - a 1 ) Γ ( b 1 ) Γ ( b 2 ) Γ ( a 2 ) Γ ( a 3 ) Γ ( b 1 - a 1 ) Γ ( b 2 - a 1 ) c a 1 - α 3 F 3 ( a 1 , a 1 - b 1 + 1 , a 1 - b 2 + 1 ; - α + a 1 + 1 , a 1 - a 2 + 1 , a 1 - a 3 + 1 ; c ) 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[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", 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[RowBox[List["-", "\[Alpha]"]], "+", SubscriptBox["a", "1"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", 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["c", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] + Γ ( α - a 2 ) Γ ( a 1 - a 2 ) Γ ( a 3 - a 2 ) Γ ( b 1 ) Γ ( b 2 ) Γ ( a 1 ) Γ ( a 3 ) Γ ( b 1 - a 2 ) Γ ( b 2 - a 2 ) c a 2 - α 3 F 3 ( a 2 , a 2 - b 1 + 1 , a 2 - b 2 + 1 ; - α + a 2 + 1 , - a 1 + a 2 + 1 , a 2 - a 3 + 1 ; c ) 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[SubscriptBox["a", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", 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["-", "\[Alpha]"]], "+", SubscriptBox["a", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", 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["c", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] + Γ ( α - 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 ( b 1 - a 3 ) Γ ( b 2 - a 3 ) c a 3 - α 3 F 3 ( a 3 , a 3 - b 1 + 1 , a 3 - b 2 + 1 ; - α + a 3 + 1 , - a 1 + a 3 + 1 , - a 2 + a 3 + 1 ; c ) 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[SubscriptBox["a", "3"], HypergeometricPFQ, Rule[Editable, True]], ",", 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["-", "\[Alpha]"]], "+", SubscriptBox["a", "3"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", 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["c", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] /; Re ( α ) > 0 Re ( c ) > 0 Condition t 0 t α -1 -1 c t HypergeometricPFQ Subscript a 1 Subscript a 2 Subscript a 3 Subscript b 1 Subscript b 2 -1 t Gamma α Gamma Subscript a 1 -1 α Gamma Subscript a 2 -1 α Gamma Subscript a 3 -1 α Gamma Subscript b 1 Γ Subscript b 2 Gamma Subscript a 1 Gamma Subscript a 2 Gamma Subscript a 3 Gamma Subscript b 1 -1 α Gamma Subscript b 2 -1 α -1 HypergeometricPFQ α α -1 Subscript b 1 1 α -1 Subscript b 2 1 α -1 Subscript a 1 1 α -1 Subscript a 2 1 α -1 Subscript a 3 1 c Γ α -1 Subscript a 1 Gamma Subscript a 2 -1 Subscript a 1 Gamma Subscript a 3 -1 Subscript a 1 Gamma Subscript b 1 Γ Subscript b 2 Gamma Subscript a 2 Gamma Subscript a 3 Gamma Subscript b 1 -1 Subscript a 1 Gamma Subscript b 2 -1 Subscript a 1 -1 c Subscript a 1 -1 α HypergeometricPFQ Subscript a 1 Subscript a 1 -1 Subscript b 1 1 Subscript a 1 -1 Subscript b 2 1 -1 α Subscript a 1 1 Subscript a 1 -1 Subscript a 2 1 Subscript a 1 -1 Subscript a 3 1 c Γ α -1 Subscript a 2 Gamma Subscript a 1 -1 Subscript a 2 Gamma Subscript a 3 -1 Subscript a 2 Gamma Subscript b 1 Γ Subscript b 2 Gamma Subscript a 1 Gamma Subscript a 3 Gamma Subscript b 1 -1 Subscript a 2 Gamma Subscript b 2 -1 Subscript a 2 -1 c Subscript a 2 -1 α HypergeometricPFQ Subscript a 2 Subscript a 2 -1 Subscript b 1 1 Subscript a 2 -1 Subscript b 2 1 -1 α Subscript a 2 1 -1 Subscript a 1 Subscript a 2 1 Subscript a 2 -1 Subscript a 3 1 c Γ α -1 Subscript a 3 Gamma Subscript a 1 -1 Subscript a 3 Gamma Subscript a 2 -1 Subscript a 3 Gamma Subscript b 1 Γ Subscript b 2 Gamma Subscript a 1 Gamma Subscript a 2 Gamma Subscript b 1 -1 Subscript a 3 Gamma Subscript b 2 -1 Subscript a 3 -1 c Subscript a 3 -1 α HypergeometricPFQ Subscript a 3 Subscript a 3 -1 Subscript b 1 1 Subscript a 3 -1 Subscript b 2 1 -1 α Subscript a 3 1 -1 Subscript a 1 Subscript a 3 1 -1 Subscript a 2 Subscript a 3 1 c α 0 c 0 [/itex]

 Rule Form

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

 2001-10-29