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 Hypergeometric2F1

 http://functions.wolfram.com/07.23.17.0145.01

 Input Form

 Hypergeometric2F1[a, b, c, z] == Subscript[\[ScriptCapitalC], n][a, b, c, z] Hypergeometric2F1[a, b, c + n, z] - (((a - n - c) (b - n - c) z)/((n + c - 1) (n + c) (z - 1))) Subscript[\[ScriptCapitalC], n - 1][a, b, c, z] Hypergeometric2F1[a, b, c + n + 1, z] /; Subscript[\[ScriptCapitalC], 0][a, b, c, z] == 1 && Subscript[\[ScriptCapitalC], 1][a, b, c, z] == (-c + (1 - a - b + 2 c) z)/ (c (z - 1)) && Subscript[\[ScriptCapitalC], n][a, b, c, z] == ((1 - n - c + (1 - a - b + 2 (n - 1 + c)) z)/((n - 1 + c) (z - 1))) Subscript[\[ScriptCapitalC], n - 1][a, b, c, z] - (((a - c - n + 1) (b - c - n + 1) z)/((n - 2 + c) (n - 1 + c) (z - 1))) Subscript[\[ScriptCapitalC], n - 2][a, b, c, z] && Element[n, Integers] && n > 0

 Standard Form

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 MathML Form

 2 F 1 ( a , b ; c ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["a", Hypergeometric2F1, Rule[Editable, True]], ",", TagBox["b", Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox["c", Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox["z", Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] 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["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["a", Hypergeometric2F1, Rule[Editable, True]], ",", TagBox["b", Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["c", "+", "n"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox["z", Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] - ( a - n - c ) ( b - n - c ) z ( n + c - 1 ) ( n + c ) ( z - 1 ) 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["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["a", Hypergeometric2F1, Rule[Editable, True]], ",", TagBox["b", Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["c", "+", "n", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox["z", Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] /; 𝒞 0 ( a , b , c , z ) 1 𝒞 1 ( a , b , c , z ) ( 1 - a - b + 2 c ) z - c c ( z - 1 ) 𝒞 n ( a , b , c , z ) 1 - c - n + ( 1 - a - b + 2 ( c + n - 1 ) ) z ( c + n - 1 ) ( z - 1 ) 𝒞 n - 1 ( a , b , c , z ) - ( a - c - n + 1 ) ( b - c - n + 1 ) z ( c + n - 2 ) ( c + n - 1 ) ( z - 1 ) 𝒞 n - 2 ( a , b , c , z ) n + Condition Hypergeometric2F1 a b c z Subscript 𝒞 n a b c z Hypergeometric2F1 a b c n z -1 a -1 n -1 c b -1 n -1 c z n c -1 n c z -1 -1 Subscript 𝒞 n -1 a b c z Hypergeometric2F1 a b c n 1 z Subscript 𝒞 0 a b c z 1 Subscript 𝒞 1 a b c z 1 -1 a -1 b 2 c z -1 c c z -1 -1 Subscript 𝒞 n a b c z 1 -1 c -1 n 1 -1 a -1 b 2 c n -1 z c n -1 z -1 -1 Subscript 𝒞 n -1 a b c z -1 a -1 c -1 n 1 b -1 c -1 n 1 z c n -2 c n -1 z -1 -1 Subscript 𝒞 n -2 a b c z n SuperPlus [/itex]

 Rule Form

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

 2007-05-02