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 HypergeometricPFQ

 http://functions.wolfram.com/07.25.06.0014.01

 Input Form

 HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2]}, {Subscript[b, 1], Subscript[b, 2]}, z] \[Proportional] HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2]}, {Subscript[b, 1], Subscript[b, 2]}, Subscript[z, 0]] + ((Subscript[a, 1] Subscript[a, 2])/(Subscript[b, 1] Subscript[b, 2])) HypergeometricPFQ[{1 + Subscript[a, 1], 1 + Subscript[a, 2]}, {1 + Subscript[b, 1], 1 + Subscript[b, 2]}, Subscript[z, 0]] (z - Subscript[z, 0]) + ((Subscript[a, 1] (1 + Subscript[a, 1]) Subscript[a, 2] (1 + Subscript[a, 2]))/(2 Subscript[b, 1] (1 + Subscript[b, 1]) Subscript[b, 2] (1 + Subscript[b, 2]))) HypergeometricPFQ[ {2 + Subscript[a, 1], 2 + Subscript[a, 2]}, {2 + Subscript[b, 1], 2 + Subscript[b, 2]}, Subscript[z, 0]] (z - Subscript[z, 0])^2 + O[(z - Subscript[z, 0])^3]

 Standard Form

 Cell[BoxData[RowBox[List[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"]], "]"]], "\[Proportional]", RowBox[List[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"]]], "}"]], ",", SubscriptBox["z", "0"]]], "]"]], "+", RowBox[List[FractionBox[RowBox[List[SubscriptBox["a", "1"], " ", SubscriptBox["a", "2"], " "]], RowBox[List[SubscriptBox["b", "1"], " ", SubscriptBox["b", "2"]]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "1"]]], ",", RowBox[List["1", "+", SubscriptBox["a", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["b", "1"]]], ",", RowBox[List["1", "+", SubscriptBox["b", "2"]]]]], "}"]], ",", SubscriptBox["z", "0"]]], "]"]], " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List[SubscriptBox["a", "1"], " ", RowBox[List["(", RowBox[List["1", "+", SubscriptBox["a", "1"]]], ")"]], " ", SubscriptBox["a", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SubscriptBox["a", "2"]]], ")"]], " "]], RowBox[List["2", " ", SubscriptBox["b", "1"], " ", RowBox[List["(", RowBox[List["1", "+", SubscriptBox["b", "1"]]], ")"]], " ", SubscriptBox["b", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SubscriptBox["b", "2"]]], ")"]]]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["2", "+", SubscriptBox["a", "1"]]], ",", RowBox[List["2", "+", SubscriptBox["a", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", SubscriptBox["b", "1"]]], ",", RowBox[List["2", "+", SubscriptBox["b", "2"]]]]], "}"]], ",", SubscriptBox["z", "0"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "2"]]], "+", RowBox[List["O", "[", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "3"], "]"]]]]]]]]

 MathML Form

 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] 2 F 2 ( a 1 , a 2 ; b 1 , b 2 ; z 0 ) 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[SubscriptBox["z", "0"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] + a 1 a 2 b 1 b 2 2 F 2 ( a 1 + 1 , a 2 + 1 ; b 1 + 1 , b 2 + 1 ; z 0 ) 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"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["b", "1"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["b", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[SubscriptBox["z", "0"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ( z - z 0 ) + a 1 ( a 1 + 1 ) a 2 ( a 2 + 1 ) 2 b 1 ( b 1 + 1 ) b 2 ( b 2 + 1 ) 2 F 2 ( a 1 + 2 , a 2 + 2 ; b 1 + 2 , b 2 + 2 ; z 0 ) 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"], "+", "2"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "2"], "+", "2"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["b", "1"], "+", "2"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["b", "2"], "+", "2"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[SubscriptBox["z", "0"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ( z - z 0 ) 2 + O ( ( z - z 0 ) 3 ) Proportional HypergeometricPFQ Subscript a 1 Subscript a 2 Subscript b 1 Subscript b 2 z HypergeometricPFQ Subscript a 1 Subscript a 2 Subscript b 1 Subscript b 2 Subscript z 0 Subscript a 1 Subscript a 2 Subscript b 1 Subscript b 2 -1 HypergeometricPFQ Subscript a 1 1 Subscript a 2 1 Subscript b 1 1 Subscript b 2 1 Subscript z 0 z -1 Subscript z 0 Subscript a 1 Subscript a 1 1 Subscript a 2 Subscript a 2 1 2 Subscript b 1 Subscript b 1 1 Subscript b 2 Subscript b 2 1 -1 HypergeometricPFQ Subscript a 1 2 Subscript a 2 2 Subscript b 1 2 Subscript b 2 2 Subscript z 0 z -1 Subscript z 0 2 O z -1 Subscript z 0 3 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", 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_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["aa", "1"], ",", SubscriptBox["aa", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["bb", "1"], ",", SubscriptBox["bb", "2"]]], "}"]], ",", SubscriptBox["zz", "0"]]], "]"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["aa", "1"], " ", SubscriptBox["aa", "2"]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["aa", "1"]]], ",", RowBox[List["1", "+", SubscriptBox["aa", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["bb", "1"]]], ",", RowBox[List["1", "+", SubscriptBox["bb", "2"]]]]], "}"]], ",", SubscriptBox["zz", "0"]]], "]"]], " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]]]], RowBox[List[SubscriptBox["bb", "1"], " ", SubscriptBox["bb", "2"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["aa", "1"], " ", RowBox[List["(", RowBox[List["1", "+", SubscriptBox["aa", "1"]]], ")"]], " ", SubscriptBox["aa", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SubscriptBox["aa", "2"]]], ")"]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["2", "+", SubscriptBox["aa", "1"]]], ",", RowBox[List["2", "+", SubscriptBox["aa", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", SubscriptBox["bb", "1"]]], ",", RowBox[List["2", "+", SubscriptBox["bb", "2"]]]]], "}"]], ",", SubscriptBox["zz", "0"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "2"]]], RowBox[List["2", " ", SubscriptBox["bb", "1"], " ", RowBox[List["(", RowBox[List["1", "+", SubscriptBox["bb", "1"]]], ")"]], " ", SubscriptBox["bb", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SubscriptBox["bb", "2"]]], ")"]]]]], "+", SuperscriptBox[RowBox[List["O", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], "3"]]]]]]]

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

 2007-05-02