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 HypergeometricPFQ

 http://functions.wolfram.com/07.22.17.0004.01

 Input Form

 HypergeometricPFQ[{Subscript[a, 1]}, {b, Subscript[b, 2]}, z] == ((Subscript[B, 1] + z Subscript[C, 1])/z) HypergeometricPFQ[ {Subscript[a, 1]}, {b - 1, Subscript[b, 2]}, z] + (Subscript[B, 2]/z) HypergeometricPFQ[{Subscript[a, 1]}, {b - 2, Subscript[b, 2]}, z] + (Subscript[B, 3]/z) HypergeometricPFQ[{Subscript[a, 1]}, {b - 3, Subscript[b, 2]}, z] /; Subscript[B, 1] == ((b - 1) (b - 2) (1 - b + Subscript[b, 2]))/ (b - Subscript[a, 1] - 1) && Subscript[C, 1] == (b - 1)/(b - Subscript[a, 1] - 1) && Subscript[B, 2] == ((b - 1) (b - 2) (2 b - Subscript[b, 2] - 4))/(b - Subscript[a, 1] - 1) && Subscript[B, 3] == -(((b - 1) (b - 2) (b - 3))/(b - Subscript[a, 1] - 1))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", SubscriptBox["a", "1"], "}"]], ",", RowBox[List["{", RowBox[List["b", ",", SubscriptBox["b", "2"]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[" ", RowBox[List[SubscriptBox["B", "1"], "+", RowBox[List["z", " ", SubscriptBox["C", "1"]]]]]]], "z"], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", SubscriptBox["a", "1"], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["b", "-", "1"]], ",", SubscriptBox["b", "2"]]], "}"]], ",", "z"]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List[" ", SubscriptBox["B", "2"]]], "z"], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", SubscriptBox["a", "1"], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["b", "-", "2"]], ",", SubscriptBox["b", "2"]]], "}"]], ",", "z"]], "]"]]]], "+", RowBox[List[FractionBox[SubscriptBox["B", "3"], "z"], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", SubscriptBox["a", "1"], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["b", "-", "3"]], ",", SubscriptBox["b", "2"]]], "}"]], ",", "z"]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["B", "1"], "\[Equal]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["b", "-", "1"]], ")"]], RowBox[List["(", RowBox[List["b", "-", "2"]], ")"]], RowBox[List["(", RowBox[List["1", "-", "b", "+", SubscriptBox["b", "2"]]], ")"]]]], RowBox[List[" ", RowBox[List["b", "-", SubscriptBox["a", "1"], "-", "1"]]]]]]], "\[And]", RowBox[List[SubscriptBox["C", "1"], "\[Equal]", FractionBox[RowBox[List["b", "-", "1", " "]], RowBox[List[" ", RowBox[List["b", "-", SubscriptBox["a", "1"], "-", "1"]]]]]]], "\[And]", RowBox[List[SubscriptBox["B", "2"], "\[Equal]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["b", "-", "1"]], ")"]], RowBox[List["(", RowBox[List["b", "-", "2"]], ")"]], RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b"]], "-", SubscriptBox["b", "2"], "-", "4"]], ")"]]]], RowBox[List[" ", RowBox[List["b", "-", SubscriptBox["a", "1"], "-", "1"]]]]]]], "\[And]", RowBox[List[SubscriptBox["B", "3"], "\[Equal]", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["b", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["b", "-", "2"]], ")"]], " ", RowBox[List["(", RowBox[List["b", "-", "3"]], ")"]]]], RowBox[List["b", "-", SubscriptBox["a", "1"], "-", "1"]]]]]]]]]]]]]

 MathML Form

 1 F 2 ( a 1 ; b , b 2 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[SubscriptBox["a", "1"], 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 + z C 1 z 1 F 2 ( a 1 ; b - 1 , b 2 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[SubscriptBox["a", "1"], 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 z 1 F 2 ( a 1 ; b - 2 , b 2 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[SubscriptBox["a", "1"], 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] + B 3 z 1 F 2 ( a 1 ; b - 3 , b 2 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[SubscriptBox["a", "1"], 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 - 1 ) ( b - 2 ) ( b 2 - b + 1 ) b - a 1 - 1 C 1 b - 1 b - a 1 - 1 B 2 ( b - 1 ) ( b - 2 ) ( 2 b - b 2 - 4 ) b - a 1 - 1 B 3 - ( b - 1 ) ( b - 2 ) ( b - 3 ) b - a 1 - 1 Condition HypergeometricPFQ Subscript a 1 b Subscript b 2 z Subscript B 1 z Subscript C 1 z -1 HypergeometricPFQ Subscript a 1 b -1 Subscript b 2 z Subscript B 2 z -1 HypergeometricPFQ Subscript a 1 b -2 Subscript b 2 z Subscript B 3 z -1 HypergeometricPFQ Subscript a 1 b -3 Subscript b 2 z Subscript B 1 b -1 b -2 Subscript b 2 -1 b 1 b -1 Subscript a 1 -1 -1 Subscript C 1 b -1 b -1 Subscript a 1 -1 -1 Subscript B 2 b -1 b -2 2 b -1 Subscript b 2 -4 b -1 Subscript a 1 -1 -1 Subscript B 3 -1 b -1 b -2 b -3 b -1 Subscript a 1 -1 -1 [/itex]

 Rule Form

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

 2001-10-29