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 Hypergeometric2F1Regularized

 http://functions.wolfram.com/07.24.17.0142.01

 Input Form

 Hypergeometric2F1Regularized[a, b, c, z] == ((1 - z)/((n + a - c) (n + b - c) z)) Subscript[\[ScriptCapitalC], n - 1][ a, b, c, z] Hypergeometric2F1Regularized[a, b, c - n - 1, z] + Subscript[\[ScriptCapitalC], n][a, b, c, z] Hypergeometric2F1Regularized[ a, b, c - n, z] /; Subscript[\[ScriptCapitalC], 0][a, b, c, z] == 1 && Subscript[\[ScriptCapitalC], 1][a, b, c, z] == (2 - c - (3 + a + b - 2 c) z)/((1 + a - c) (1 + b - c) z) && Subscript[\[ScriptCapitalC], n][a, b, c, z] == ((n + 1 - c - (1 + a + b + 2 (n - c)) z)/((n + a - c) (n + b - c) z)) Subscript[\[ScriptCapitalC], n - 1][a, b, c, z] + ((1 - z)/((n - 1 + a - c) (n - 1 + b - c) z)) Subscript[\[ScriptCapitalC], n - 2][a, b, c, z] && Element[n, Integers] && n > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["a", ",", "b", ",", "c", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List["1", "-", "z"]], RowBox[List[RowBox[List["(", RowBox[List["n", "+", "a", "-", "c"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "+", "b", "-", "c"]], ")"]], " ", "z"]]], RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "1"]]], "[", RowBox[List["a", ",", "b", ",", "c", ",", "z"]], "]"]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["a", ",", "b", ",", RowBox[List["c", "-", "n", "-", "1"]], ",", "z"]], "]"]]]], "+", RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "n"], "[", RowBox[List["a", ",", "b", ",", "c", ",", "z"]], "]"]], RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["a", ",", "b", ",", RowBox[List["c", "-", "n"]], ",", "z"]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "0"], "[", RowBox[List["a", ",", "b", ",", "c", ",", "z"]], "]"]], "\[Equal]", "1"]], "\[And]", RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "1"], "[", RowBox[List["a", ",", "b", ",", "c", ",", "z"]], "]"]], "\[Equal]", FractionBox[RowBox[List["2", "-", "c", "-", RowBox[List[RowBox[List["(", RowBox[List["3", "+", "a", "+", "b", "-", RowBox[List["2", " ", "c"]]]], ")"]], " ", "z"]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", "a", "-", "c"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", "b", "-", "c"]], ")"]], " ", "z"]]]]], " ", "\[And]", RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "n"], "[", RowBox[List["a", ",", "b", ",", "c", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List["(", RowBox[List["n", "+", "1", "-", "c", "-", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "a", "+", "b", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["n", "-", "c"]], ")"]]]]]], ")"]], " ", "z"]]]], ")"]], RowBox[List[RowBox[List["(", RowBox[List["n", "+", "a", "-", "c"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "+", "b", "-", "c"]], ")"]], " ", "z"]]], RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "1"]]], "[", RowBox[List["a", ",", "b", ",", "c", ",", "z"]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], " "]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1", "+", "a", "-", "c"]], ")"]], RowBox[List["(", RowBox[List["n", "-", "1", "+", "b", "-", "c"]], ")"]], " ", "z"]]], RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "2"]]], "[", RowBox[List["a", ",", "b", ",", "c", ",", "z"]], "]"]]]]]]]], "\[And]", RowBox[List["Element", "[", RowBox[List["n", ",", "Integers"]], "]"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]

 MathML Form

 2 F ~ 1 ( a , b ; c ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["a", Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox["b", Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox["c", Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox["z", Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] 1 - z ( a - c + n ) ( b - c + n ) z 𝒞 n - 1 ( a , b , c , z ) 2 F ~ 1 ( a , b ; c - n - 1 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["a", Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox["b", Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["c", "-", "n", "-", "1"]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox["z", Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] + 𝒞 n ( a , b , c , z ) 2 F ~ 1 ( a , b ; c - n ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["a", Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox["b", Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["c", "-", "n"]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox["z", Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] /; 𝒞 0 ( a , b , c , z ) 1 𝒞 1 ( a , b , c , z ) 2 - c - ( a + b - 2 c + 3 ) z ( a - c + 1 ) ( b - c + 1 ) z 𝒞 n ( a , b , c , z ) 1 - z ( a - c + n - 1 ) ( b - c + n - 1 ) z 𝒞 n - 2 ( a , b , c , z ) + 1 - c + n - ( a + b + 2 ( n - c ) + 1 ) z ( a - c + n ) ( b - c + n ) z 𝒞 n - 1 ( a , b , c , z ) n + Condition Hypergeometric2F1Regularized a b c z 1 -1 z a -1 c n b -1 c n z -1 Subscript 𝒞 n -1 a b c z Hypergeometric2F1Regularized a b c -1 n -1 z Subscript 𝒞 n a b c z Hypergeometric2F1Regularized a b c -1 n z Subscript 𝒞 0 a b c z 1 Subscript 𝒞 1 a b c z 2 -1 c -1 a b -1 2 c 3 z a -1 c 1 b -1 c 1 z -1 Subscript 𝒞 n a b c z 1 -1 z a -1 c n -1 b -1 c n -1 z -1 Subscript 𝒞 n -2 a b c z 1 -1 c n -1 a b 2 n -1 c 1 z a -1 c n b -1 c n z -1 Subscript 𝒞 n -1 a b c z n SuperPlus [/itex]

 Rule Form

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

 2007-05-02