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 HypergeometricPFQ

 http://functions.wolfram.com/07.25.13.0001.01

 Input Form

 z^2 Derivative[3][w][z] + (Subscript[b, 1] + Subscript[b, 2] + 1 - z) z Derivative[2][w][z] + (Subscript[b, 1] Subscript[b, 2] - z (Subscript[a, 1] + Subscript[a, 2] + 1)) Derivative[1][w][z] - Subscript[a, 1] Subscript[a, 2] w[z] == 0 /; w[z] == Subscript[c, 1] HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2]}, {Subscript[b, 1], Subscript[b, 2]}, z] + Subscript[c, 2] z^(1 - Subscript[b, 1]) HypergeometricPFQ[ {1 + Subscript[a, 1] - Subscript[b, 1], 1 + Subscript[a, 2] - Subscript[b, 1]}, {2 - Subscript[b, 1], 1 - Subscript[b, 1] + Subscript[b, 2]}, z] + Subscript[c, 3] z^(1 - Subscript[b, 2]) HypergeometricPFQ[{1 + Subscript[a, 1] - Subscript[b, 2], 1 + Subscript[a, 2] - Subscript[b, 2]}, {2 - Subscript[b, 2], 1 + Subscript[b, 1] - Subscript[b, 2]}, z] && !Element[Subscript[b, 1], Integers] && !Element[Subscript[b, 2], Integers] && !Element[Subscript[b, 1] - Subscript[b, 2], Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List[SuperscriptBox["w", TagBox[RowBox[List["(", "3", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "z", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["b", "1"], "+", SubscriptBox["b", "2"], "+", "1", "-", "z"]], ")"]], " ", "z", " ", RowBox[List[SuperscriptBox["w", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["b", "1"], " ", SubscriptBox["b", "2"]]], "-", RowBox[List["z", " ", RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], "+", SubscriptBox["a", "2"], "+", "1"]], ")"]]]]]], ")"]], " ", RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], "-", RowBox[List[SubscriptBox["a", "1"], " ", SubscriptBox["a", "2"], " ", RowBox[List["w", "[", "z", "]"]]]]]], "\[Equal]", "0"]], "/;", RowBox[List[RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["c", "1"], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"]]], "}"]], ",", "z"]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], SuperscriptBox["z", RowBox[List["1", "-", SubscriptBox["b", "1"]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "1"], "-", SubscriptBox["b", "1"]]], ",", RowBox[List["1", "+", SubscriptBox["a", "2"], "-", SubscriptBox["b", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "-", SubscriptBox["b", "1"]]], ",", RowBox[List["1", "-", SubscriptBox["b", "1"], "+", SubscriptBox["b", "2"]]]]], "}"]], ",", "z"]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "3"], SuperscriptBox["z", RowBox[List["1", "-", SubscriptBox["b", "2"]]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "1"], "-", SubscriptBox["b", "2"]]], ",", RowBox[List["1", "+", SubscriptBox["a", "2"], "-", SubscriptBox["b", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "-", SubscriptBox["b", "2"]]], ",", RowBox[List["1", "+", SubscriptBox["b", "1"], "-", SubscriptBox["b", "2"]]]]], "}"]], ",", "z"]], "]"]]]]]]]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List[SubscriptBox["b", "1"], ",", "Integers"]], "]"]], "]"]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List[SubscriptBox["b", "2"], ",", "Integers"]], "]"]], "]"]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List[RowBox[List[SubscriptBox["b", "1"], "-", SubscriptBox["b", "2"]]], ",", "Integers"]], "]"]], "]"]]]]]]]]

 MathML Form

 z 2 w ( 3 ) TagBox[RowBox[List["(", "3", ")"]], Derivative] ( z ) + z ( 1 - z + b 1 + b 2 ) w ′′ ( z ) + ( b 1 b 2 - z ( a 1 + a 2 + 1 ) ) w ( z ) - a 1 a 2 w ( z ) 0 /; w ( z ) c 1 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[SubscriptBox["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] + c 2 z 1 - b 1 2 F 2 ( a 1 - b 1 + 1 , a 2 - b 1 + 1 ; 2 - b 1 , - b 1 + b 2 + 1 ; 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[SubscriptBox["a", "1"], "-", SubscriptBox["b", "1"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "2"], "-", SubscriptBox["b", "1"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["2", "-", SubscriptBox["b", "1"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", SubscriptBox["b", "1"]]], "+", SubscriptBox["b", "2"], "+", "1"]], 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 1 - b 2 2 F 2 ( a 1 - b 2 + 1 , a 2 - b 2 + 1 ; 2 - b 2 , b 1 - b 2 + 1 ; 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[SubscriptBox["a", "1"], "-", SubscriptBox["b", "2"], "+", "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["2", "-", SubscriptBox["b", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["b", "1"], "-", SubscriptBox["b", "2"], "+", "1"]], 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 TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] b 2 TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] b 1 - b 2 TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition z 2 z 3 w z z 1 -1 z Subscript b 1 Subscript b 2 z 2 w z Subscript b 1 Subscript b 2 -1 z Subscript a 1 Subscript a 2 1 z w z -1 Subscript a 1 Subscript a 2 w z 0 w z Subscript c 1 HypergeometricPFQ Subscript a 1 Subscript a 2 Subscript b 1 Subscript b 2 z Subscript c 2 z 1 -1 Subscript b 1 HypergeometricPFQ Subscript a 1 -1 Subscript b 1 1 Subscript a 2 -1 Subscript b 1 1 2 -1 Subscript b 1 -1 Subscript b 1 Subscript b 2 1 z Subscript c 3 z 1 -1 Subscript b 2 HypergeometricPFQ Subscript a 1 -1 Subscript b 2 1 Subscript a 2 -1 Subscript b 2 1 2 -1 Subscript b 2 Subscript b 1 -1 Subscript b 2 1 z Subscript b 1 Subscript b 2 Subscript b 1 -1 Subscript b 2 [/itex]

 Rule Form

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

 2001-10-29