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 HypergeometricPFQ

 http://functions.wolfram.com/07.27.17.0003.01

 Input Form

 HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3]}, {b, Subscript[b, 2]}, z] == ((Subscript[B, 1] + Subscript[C, 1] z)/(z - 1)) HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3]}, {b + 1, Subscript[b, 2]}, z] + ((Subscript[B, 2] + Subscript[C, 2] z)/ (z - 1)) HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3]}, {b + 2, Subscript[b, 2]}, z] + ((Subscript[C, 3] z)/(z - 1)) HypergeometricPFQ[ {Subscript[a, 1], Subscript[a, 2], Subscript[a, 3]}, {b + 3, Subscript[b, 2]}, z] /; Subscript[B, 1] == (Subscript[b, 2] - 2 - 2 b)/b && Subscript[C, 1] == (3 - Subscript[a, 1] - Subscript[a, 2] - Subscript[a, 3] + 3 b)/b && Subscript[B, 2] == (b - Subscript[b, 2] + 2)/b && Subscript[C, 2] == ((3 + 2 b) (Subscript[a, 1] + Subscript[a, 2] + Subscript[a, 3]) - 7 - 9 b - 3 b^2 - Subscript[a, 1] Subscript[a, 2] - Subscript[a, 1] Subscript[a, 3] - Subscript[a, 2] Subscript[a, 3])/ (b (1 + b)) && Subscript[C, 3] == ((2 - Subscript[a, 1] + b) (2 - Subscript[a, 2] + b) (2 - Subscript[a, 3] + b))/(b (1 + b) (2 + b))

 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"]]], "}"]], ",", RowBox[List["{", RowBox[List["b", ",", SubscriptBox["b", "2"]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[" ", RowBox[List[SubscriptBox["B", "1"], "+", RowBox[List[SubscriptBox["C", "1"], "z"]]]]]], RowBox[List["z", "-", "1"]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", SubscriptBox["a", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["b", "+", "1"]], ",", SubscriptBox["b", "2"]]], "}"]], ",", "z"]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List[" ", RowBox[List[SubscriptBox["B", "2"], "+", RowBox[List[SubscriptBox["C", "2"], "z"]]]]]], RowBox[List["z", "-", "1"]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", SubscriptBox["a", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["b", "+", "2"]], ",", SubscriptBox["b", "2"]]], "}"]], ",", "z"]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List[" ", RowBox[List[SubscriptBox["C", "3"], "z"]]]], RowBox[List["z", "-", "1"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", SubscriptBox["a", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["b", "+", "3"]], ",", SubscriptBox["b", "2"]]], "}"]], ",", "z"]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["B", "1"], "\[Equal]", FractionBox[RowBox[List[SubscriptBox["b", "2"], "-", "2", "-", RowBox[List["2", " ", "b"]]]], "b"]]], "\[And]", RowBox[List[SubscriptBox["C", "1"], "\[Equal]", FractionBox[RowBox[List["3", "-", SubscriptBox["a", "1"], "-", SubscriptBox["a", "2"], "-", SubscriptBox["a", "3"], "+", RowBox[List["3", " ", "b"]]]], "b"]]], "\[And]", RowBox[List[SubscriptBox["B", "2"], "\[Equal]", FractionBox[RowBox[List["b", "-", SubscriptBox["b", "2"], "+", "2"]], "b"]]], "\[And]", RowBox[List[SubscriptBox["C", "2"], "\[Equal]", FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], "+", SubscriptBox["a", "2"], "+", SubscriptBox["a", "3"]]], ")"]]]], "-", "7", "-", RowBox[List["9", " ", "b"]], "-", RowBox[List["3", " ", SuperscriptBox["b", "2"]]], "-", RowBox[List[SubscriptBox["a", "1"], " ", SubscriptBox["a", "2"]]], "-", RowBox[List[SubscriptBox["a", "1"], " ", SubscriptBox["a", "3"]]], "-", RowBox[List[SubscriptBox["a", "2"], " ", SubscriptBox["a", "3"]]]]], RowBox[List["b", " ", RowBox[List["(", RowBox[List["1", "+", "b"]], ")"]]]]]]], "\[And]", RowBox[List[SubscriptBox["C", "3"], "\[Equal]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", "-", SubscriptBox["a", "1"], "+", "b"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "-", SubscriptBox["a", "2"], "+", "b"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "-", SubscriptBox["a", "3"], "+", "b"]], ")"]]]], RowBox[List["b", " ", RowBox[List["(", RowBox[List["1", "+", "b"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "b"]], ")"]]]]]]]]]]]]]

 MathML Form

 3 F 2 ( a 1 , a 2 , a 3 ; b , 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["b", 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] B 1 + C 1 z z - 1 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[RowBox[List["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] + B 2 + C 2 z z - 1 3 F 2 ( a 1 , a 2 , a 3 ; b + 2 , 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[RowBox[List["b", "+", "2"]], 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] + C 3 z z - 1 3 F 2 ( a 1 , a 2 , a 3 ; b + 3 , 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[RowBox[List["b", "+", "3"]], 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] /; B 1 b 2 - 2 b - 2 b C 1 3 b - a 1 - a 2 - a 3 + 3 b B 2 b - b 2 + 2 b C 2 ( 2 b + 3 ) ( a 1 + a 2 + a 3 ) - 3 b 2 - 9 b - a 1 a 2 - a 1 a 3 - a 2 a 3 - 7 b ( b + 1 ) C 3 ( b - a 1 + 2 ) ( b - a 2 + 2 ) ( b - a 3 + 2 ) b ( b + 1 ) ( b + 2 ) Condition HypergeometricPFQ Subscript a 1 Subscript a 2 Subscript a 3 b Subscript b 2 z Subscript B 1 Subscript C 1 z z -1 -1 HypergeometricPFQ Subscript a 1 Subscript a 2 Subscript a 3 b 1 Subscript b 2 z Subscript B 2 Subscript C 2 z z -1 -1 HypergeometricPFQ Subscript a 1 Subscript a 2 Subscript a 3 b 2 Subscript b 2 z Subscript C 3 z z -1 -1 HypergeometricPFQ Subscript a 1 Subscript a 2 Subscript a 3 b 3 Subscript b 2 z Subscript B 1 Subscript b 2 -1 2 b -2 b -1 Subscript C 1 3 b -1 Subscript a 1 -1 Subscript a 2 -1 Subscript a 3 3 b -1 Subscript B 2 b -1 Subscript b 2 2 b -1 Subscript C 2 2 b 3 Subscript a 1 Subscript a 2 Subscript a 3 -1 3 b 2 -1 9 b -1 Subscript a 1 Subscript a 2 -1 Subscript a 1 Subscript a 3 -1 Subscript a 2 Subscript a 3 -7 b b 1 -1 Subscript C 3 b -1 Subscript a 1 2 b -1 Subscript a 2 2 b -1 Subscript a 3 2 b b 1 b 2 -1 [/itex]

 Rule Form

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

 2001-10-29