html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 Hypergeometric2F1Regularized

 http://functions.wolfram.com/07.24.17.0140.01

 Input Form

 Hypergeometric2F1Regularized[a, b, c, z] == Subscript[\[ScriptCapitalC], n][a, b, c, z] Hypergeometric2F1Regularized[a, b - n, c, z] + ((n - b + c)/((n - b) (z - 1))) Subscript[\[ScriptCapitalC], n - 1][a, b, c, z] Hypergeometric2F1Regularized[a, b - n - 1, c, z] /; Subscript[\[ScriptCapitalC], 0][a, b, c, z] == 1 && Subscript[\[ScriptCapitalC], 1][a, b, c, z] == (2 - 2 b + c + (b - a - 1) z)/((b - 1) (z - 1)) && Subscript[\[ScriptCapitalC], n][a, b, c, z] == (-((2 n - 2 b + c + (b - a - n) z)/((n - b) (z - 1)))) Subscript[\[ScriptCapitalC], n - 1][a, b, c, z] + ((n - 1 - b + c)/((n - 1 - b) (z - 1))) 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[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "n"], "[", RowBox[List["a", ",", "b", ",", "c", ",", "z"]], "]"]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["a", ",", RowBox[List["b", "-", "n"]], ",", "c", ",", "z"]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List["n", "-", "b", "+", "c"]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "b"]], ")"]], " ", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]]], " ", RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "1"]]], "[", RowBox[List["a", ",", "b", ",", "c", ",", "z"]], "]"]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["a", ",", RowBox[List["b", "-", "n", "-", "1"]], ",", "c", ",", "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", "-", RowBox[List["2", " ", "b"]], "+", "c", "+", RowBox[List[RowBox[List["(", RowBox[List["b", "-", "a", "-", "1"]], ")"]], " ", "z"]]]], RowBox[List[RowBox[List["(", RowBox[List["b", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]]]]], " ", "\[And]", RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "n"], "[", RowBox[List["a", ",", "b", ",", "c", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["2", "n"]], "-", RowBox[List["2", "b"]], "+", "c", "+", RowBox[List[RowBox[List["(", RowBox[List["b", "-", "a", "-", "n"]], ")"]], " ", "z"]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "b"]], ")"]], " ", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]]]]], RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "1"]]], "[", RowBox[List["a", ",", "b", ",", "c", ",", "z"]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List["n", "-", "1", "-", "b", "+", "c", " "]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1", "-", "b"]], ")"]], RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]]], 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] n - b + c ( n - b ) ( z - 1 ) 𝒞 n - 1 ( a , b , c , z ) 2 F ~ 1 ( a , b - n - 1 ; 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[RowBox[List["b", "-", "n", "-", "1"]], 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] + 𝒞 n ( a , b , c , z ) 2 F ~ 1 ( a , b - n ; 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[RowBox[List["b", "-", "n"]], 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] /; 𝒞 0 ( a , b , c , z ) 1 𝒞 1 ( a , b , c , z ) 2 - 2 b + c + ( b - a - 1 ) z ( b - 1 ) ( z - 1 ) 𝒞 n ( a , b , c , z ) - - 2 b + c + 2 n + ( b - a - n ) z ( n - b ) ( z - 1 ) 𝒞 n - 1 ( a , b , c , z ) + n - b + c - 1 ( n - b - 1 ) ( z - 1 ) 𝒞 n - 2 ( a , b , c , z ) n + Condition Hypergeometric2F1Regularized a b c z n -1 b c n -1 b z -1 -1 Subscript 𝒞 n -1 a b c z Hypergeometric2F1Regularized a b -1 n -1 c z Subscript 𝒞 n a b c z Hypergeometric2F1Regularized a b -1 n c z Subscript 𝒞 0 a b c z 1 Subscript 𝒞 1 a b c z 2 -1 2 b c b -1 a -1 z b -1 z -1 -1 Subscript 𝒞 n a b c z -1 -2 b c 2 n b -1 a -1 n z n -1 b z -1 -1 Subscript 𝒞 n -1 a b c z n -1 b c -1 n -1 b -1 z -1 -1 Subscript 𝒞 n -2 a b c z n SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["a_", ",", "b_", ",", "c_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "n"], "[", RowBox[List["a", ",", "b", ",", "c", ",", "z"]], "]"]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["a", ",", RowBox[List["b", "-", "n"]], ",", "c", ",", "z"]], "]"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "b", "+", "c"]], ")"]], " ", RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "1"]]], "[", RowBox[List["a", ",", "b", ",", "c", ",", "z"]], "]"]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["a", ",", RowBox[List["b", "-", "n", "-", "1"]], ",", "c", ",", "z"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "b"]], ")"]], " ", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "0"], "[", RowBox[List["a", ",", "b", ",", "c", ",", "z"]], "]"]], "\[Equal]", "1"]], "&&", RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "1"], "[", RowBox[List["a", ",", "b", ",", "c", ",", "z"]], "]"]], "\[Equal]", FractionBox[RowBox[List["2", "-", RowBox[List["2", " ", "b"]], "+", "c", "+", RowBox[List[RowBox[List["(", RowBox[List["b", "-", "a", "-", "1"]], ")"]], " ", "z"]]]], RowBox[List[RowBox[List["(", RowBox[List["b", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]]]]], "&&", RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "n"], "[", RowBox[List["a", ",", "b", ",", "c", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "-", RowBox[List["2", " ", "b"]], "+", "c", "+", RowBox[List[RowBox[List["(", RowBox[List["b", "-", "a", "-", "n"]], ")"]], " ", "z"]]]], ")"]], " ", RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "1"]]], "[", RowBox[List["a", ",", "b", ",", "c", ",", "z"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "b"]], ")"]], " ", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1", "-", "b", "+", "c"]], ")"]], " ", RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "2"]]], "[", RowBox[List["a", ",", "b", ",", "c", ",", "z"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1", "-", "b"]], ")"]], " ", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]]]]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]

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

 2007-05-02