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 HypergeometricPFQ

 http://functions.wolfram.com/07.27.06.0037.01

 Input Form

 HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3]}, {Subscript[b, 1], Subscript[b, 2]}, z] == ((Gamma[Subscript[a, 2] - Subscript[a, 1]] Gamma[Subscript[a, 3] - Subscript[a, 1]] Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]])/(Gamma[Subscript[a, 2]] Gamma[Subscript[a, 3]] Gamma[Subscript[b, 1] - Subscript[a, 1]] Gamma[Subscript[b, 2] - Subscript[a, 1]])) Sum[((Pochhammer[Subscript[a, 1], u]/((-x)^u u!)) HypergeometricPFQ[{Subscript[a, 1] + u, 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]}, 1/x] (z - x)^u)/ (E^(Pi I Subscript[a, 1] (1 + 2 Floor[Arg[x - z]/(2 Pi)])) x^Subscript[a, 1]), {u, 0, Infinity}] + ((Gamma[Subscript[a, 1] - Subscript[a, 2]] Gamma[Subscript[a, 3] - Subscript[a, 2]] Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]])/(Gamma[Subscript[a, 1]] Gamma[Subscript[a, 3]] Gamma[Subscript[b, 1] - Subscript[a, 2]] Gamma[Subscript[b, 2] - Subscript[a, 2]])) Sum[((Pochhammer[Subscript[a, 2], u]/((-x)^u u!)) HypergeometricPFQ[{Subscript[a, 2] + u, 1 + Subscript[a, 2] - Subscript[b, 1], 1 + Subscript[a, 2] - Subscript[b, 2]}, {1 + Subscript[a, 2] - Subscript[a, 1], 1 + Subscript[a, 2] - Subscript[a, 3]}, 1/x] (z - x)^u)/ (E^(Pi I Subscript[a, 2] (1 + 2 Floor[Arg[x - z]/(2 Pi)])) x^Subscript[a, 2]), {u, 0, Infinity}] + ((Gamma[Subscript[a, 1] - Subscript[a, 3]] Gamma[Subscript[a, 2] - Subscript[a, 3]] Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]])/(Gamma[Subscript[a, 1]] Gamma[Subscript[a, 2]] Gamma[Subscript[b, 1] - Subscript[a, 3]] Gamma[Subscript[b, 2] - Subscript[a, 3]])) Sum[((Pochhammer[Subscript[a, 3], u]/((-x)^u u!)) HypergeometricPFQ[{Subscript[a, 3] + u, 1 + Subscript[a, 3] - Subscript[b, 1], 1 + Subscript[a, 3] - Subscript[b, 2]}, {1 + Subscript[a, 3] - Subscript[a, 1], 1 + Subscript[a, 3] - Subscript[a, 2]}, 1/x] (z - x)^u)/ (E^(Pi I Subscript[a, 3] (1 + 2 Floor[Arg[x - z]/(2 Pi)])) x^Subscript[a, 3]), {u, 0, Infinity}] /; ForAll[{j, k}, Element[{j, k}, Integers] && j != k && 1 <= j <= 3 && 1 <= k <= 3, !Element[Subscript[a, j] - Subscript[a, k], Integers]] && x > 1

 Standard Form

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 MathML Form

 3 F 2 ( a 1 , a 2 , a 3 ; b 1 , b 2 ; z ) 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["z", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] 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 ) u = 0 ( - x ) - u ( a 1 ) u TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "1"], ")"]], "u"], Pochhammer] u ! - π a 1 ( 2 arg ( x - z ) 2 π + 1 ) x - a 1 3 F 2 ( u + a 1 , a 1 - b 1 + 1 , a 1 - b 2 + 1 ; a 1 - a 2 + 1 , a 1 - a 3 + 1 ; 1 x ) 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[RowBox[List["u", "+", 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[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[FractionBox["1", "x"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ( z - x ) u + 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 ) u = 0 ( - x ) - u ( a 2 ) u TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "2"], ")"]], "u"], Pochhammer] u ! - π a 2 ( 2 arg ( x - z ) 2 π + 1 ) x - a 2 3 F 2 ( u + a 2 , a 2 - b 1 + 1 , a 2 - b 2 + 1 ; - a 1 + a 2 + 1 , a 2 - a 3 + 1 ; 1 x ) 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[RowBox[List["u", "+", 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["-", 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[FractionBox["1", "x"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ( z - x ) u + 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 ) u = 0 ( - x ) - u ( a 3 ) u TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "3"], ")"]], "u"], Pochhammer] u ! - π a 3 ( 2 arg ( x - z ) 2 π + 1 ) x - a 3 3 F 2 ( u + a 3 , a 3 - b 1 + 1 , a 3 - b 2 + 1 ; - a 1 + a 3 + 1 , - a 2 + a 3 + 1 ; 1 x ) 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[RowBox[List["u", "+", 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["-", 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[FractionBox["1", "x"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ( z - x ) u /; { j , k } , { j , k } TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] j k 1 j 3 1 k 3 ( a j - a k TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] ) x > 1 Condition HypergeometricPFQ Subscript a 1 Subscript a 2 Subscript a 3 Subscript b 1 Subscript b 2 z Gamma Subscript a 2 -1 Subscript a 1 Gamma Subscript a 3 -1 Subscript a 1 Gamma Subscript b 1 Gamma 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 u 0 -1 x -1 u Pochhammer Subscript a 1 u u -1 -1 Subscript a 1 2 x -1 z 2 -1 1 x -1 Subscript a 1 HypergeometricPFQ u Subscript a 1 Subscript a 1 -1 Subscript b 1 1 Subscript a 1 -1 Subscript b 2 1 Subscript a 1 -1 Subscript a 2 1 Subscript a 1 -1 Subscript a 3 1 1 x -1 z -1 x u Gamma Subscript a 1 -1 Subscript a 2 Gamma Subscript a 3 -1 Subscript a 2 Gamma Subscript b 1 Gamma 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 u 0 -1 x -1 u Pochhammer Subscript a 2 u u -1 -1 Subscript a 2 2 x -1 z 2 -1 1 x -1 Subscript a 2 HypergeometricPFQ u Subscript a 2 Subscript a 2 -1 Subscript b 1 1 Subscript a 2 -1 Subscript b 2 1 -1 Subscript a 1 Subscript a 2 1 Subscript a 2 -1 Subscript a 3 1 1 x -1 z -1 x u Gamma Subscript a 1 -1 Subscript a 3 Gamma Subscript a 2 -1 Subscript a 3 Gamma Subscript b 1 Gamma 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 u 0 -1 x -1 u Pochhammer Subscript a 3 u u -1 -1 Subscript a 3 2 x -1 z 2 -1 1 x -1 Subscript a 3 HypergeometricPFQ u Subscript a 3 Subscript a 3 -1 Subscript b 1 1 Subscript a 3 -1 Subscript b 2 1 -1 Subscript a 1 Subscript a 3 1 -1 Subscript a 2 Subscript a 3 1 1 x -1 z -1 x u j k j k j k 1 j 3 1 k 3 Subscript a j -1 Subscript a k x 1 [/itex]

 Rule Form

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

 2007-05-02