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 HypergeometricPFQ

 http://functions.wolfram.com/07.25.17.0002.01

 Input Form

 HypergeometricPFQ[{a, Subscript[a, 2]}, {Subscript[b, 1], Subscript[b, 2]}, z] == (Subscript[B, 1] + Subscript[C, 1] z) HypergeometricPFQ[ {a - 1, Subscript[a, 2]}, {Subscript[b, 1], Subscript[b, 2]}, z] + (Subscript[B, 2] + Subscript[C, 2] z) HypergeometricPFQ[ {a - 2, Subscript[a, 2]}, {Subscript[b, 1], Subscript[b, 2]}, z] + Subscript[B, 3] HypergeometricPFQ[{a - 3, Subscript[a, 2]}, {Subscript[b, 1], Subscript[b, 2]}, z] /; Subscript[B, 1] == (4 - 3 a + Subscript[b, 1] + Subscript[b, 2])/(1 - a) && Subscript[C, 1] == 1/(a - 1) && Subscript[B, 2] == ((a - 2) (5 - 3 a + 2 Subscript[b, 1] + 2 Subscript[b, 2]) - Subscript[b, 1] Subscript[b, 2])/((a - 1) (a - 2)) && Subscript[C, 2] == (2 - a + Subscript[a, 2])/((a - 1) (a - 2)) && Subscript[B, 3] == ((-2 + a - Subscript[b, 1]) (-2 + a - Subscript[b, 2]))/ ((a - 1) (a - 2))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["a", ",", SubscriptBox["a", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["B", "1"], "+", RowBox[List[SubscriptBox["C", "1"], "z"]]]], ")"]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["a", "-", "1"]], ",", SubscriptBox["a", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"]]], "}"]], ",", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["B", "2"], "+", RowBox[List[SubscriptBox["C", "2"], "z"]]]], ")"]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["a", "-", "2"]], ",", SubscriptBox["a", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"]]], "}"]], ",", "z"]], "]"]]]], "+", RowBox[List[SubscriptBox["B", "3"], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["a", "-", "3"]], ",", SubscriptBox["a", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"]]], "}"]], ",", "z"]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["B", "1"], "\[Equal]", FractionBox[RowBox[List["4", "-", RowBox[List["3", " ", "a"]], "+", SubscriptBox["b", "1"], "+", SubscriptBox["b", "2"]]], RowBox[List["1", "-", "a"]]]]], "\[And]", RowBox[List[SubscriptBox["C", "1"], "\[Equal]", FractionBox["1", RowBox[List["a", "-", "1"]]]]], "\[And]", RowBox[List[SubscriptBox["B", "2"], "\[Equal]", FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "2"]], ")"]], " ", RowBox[List["(", RowBox[List["5", "-", RowBox[List["3", " ", "a"]], "+", RowBox[List["2", " ", SubscriptBox["b", "1"]]], "+", RowBox[List["2", " ", SubscriptBox["b", "2"]]]]], ")"]]]], "-", RowBox[List[SubscriptBox["b", "1"], " ", SubscriptBox["b", "2"]]]]], RowBox[List[RowBox[List["(", RowBox[List["a", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["a", "-", "2"]], ")"]]]]]]], "\[And]", RowBox[List[SubscriptBox["C", "2"], "\[Equal]", FractionBox[RowBox[List["2", "-", "a", "+", SubscriptBox["a", "2"]]], RowBox[List[RowBox[List["(", RowBox[List["a", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["a", "-", "2"]], ")"]]]]]]], "\[And]", RowBox[List[SubscriptBox["B", "3"], "\[Equal]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "a", "-", SubscriptBox["b", "1"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "a", "-", SubscriptBox["b", "2"]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["a", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["a", "-", "2"]], ")"]]]]]]]]]]]]]

 MathML Form

 2 F 2 ( a , a 2 ; b 1 , b 2 ; 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["a", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "2"], 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] ( B 1 + C 1 z ) 2 F 2 ( a - 1 , a 2 ; b 1 , b 2 ; 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["a", "-", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "2"], 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] + ( B 2 + C 2 z ) 2 F 2 ( a - 2 , a 2 ; b 1 , b 2 ; 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["a", "-", "2"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "2"], 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] + B 3 2 F 2 ( a - 3 , a 2 ; b 1 , b 2 ; 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["a", "-", "3"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "2"], 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] /; B 1 b 1 + b 2 - 3 a + 4 1 - a C 1 1 a - 1 B 2 ( a - 2 ) ( 5 - 3 a + 2 b 1 + 2 b 2 ) - b 1 b 2 ( a - 1 ) ( a - 2 ) C 2 a 2 - a + 2 ( a - 1 ) ( a - 2 ) B 3 ( a - b 1 - 2 ) ( a - b 2 - 2 ) ( a - 1 ) ( a - 2 ) Condition HypergeometricPFQ a Subscript a 2 Subscript b 1 Subscript b 2 z Subscript B 1 Subscript C 1 z HypergeometricPFQ a -1 Subscript a 2 Subscript b 1 Subscript b 2 z Subscript B 2 Subscript C 2 z HypergeometricPFQ a -2 Subscript a 2 Subscript b 1 Subscript b 2 z Subscript B 3 HypergeometricPFQ a -3 Subscript a 2 Subscript b 1 Subscript b 2 z Subscript B 1 Subscript b 1 Subscript b 2 -1 3 a 4 1 -1 a -1 Subscript C 1 1 a -1 -1 Subscript B 2 a -2 5 -1 3 a 2 Subscript b 1 2 Subscript b 2 -1 Subscript b 1 Subscript b 2 a -1 a -2 -1 Subscript C 2 Subscript a 2 -1 a 2 a -1 a -2 -1 Subscript B 3 a -1 Subscript b 1 -2 a -1 Subscript b 2 -2 a -1 a -2 -1 [/itex]

 Rule Form

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

 2001-10-29