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 JacobiP

 http://functions.wolfram.com/07.15.17.0019.01

 Input Form

 JacobiP[\[Nu], a, b, z] == Subscript[\[ScriptCapitalC], n][\[Nu], a, b, z] JacobiP[\[Nu] - n, a, b, z] + (((a - n + \[Nu]) (b - n + \[Nu]) (2 + a + b - 2 n + 2 \[Nu]))/ ((-1 + n - \[Nu]) (1 + a + b - n + \[Nu]) (a + b - 2 n + 2 \[Nu]))) Subscript[\[ScriptCapitalC], n - 1][\[Nu], a, b, z] JacobiP[\[Nu] - n - 1, a, b, z] /; Subscript[\[ScriptCapitalC], 0][\[Nu], a, b, z] == 1 && Subscript[\[ScriptCapitalC], 1][\[Nu], a, b, z] == ((-1 + a + b + 2 \[Nu]) (a^2 - b^2 + z (-2 + a + b + 2 \[Nu]) (a + b + 2 \[Nu])))/(2 \[Nu] (a + b + \[Nu]) (-2 + a + b + 2 \[Nu])) && Subscript[\[ScriptCapitalC], n][\[Nu], a, b, z] == (((1 + a + b - 2 n + 2 \[Nu]) (a^2 - b^2 + z (a + b - 2 n + 2 \[Nu]) (2 + a + b - 2 n + 2 \[Nu])))/ (2 (1 - n + \[Nu]) (1 + a + b - n + \[Nu]) (a + b - 2 n + 2 \[Nu]))) Subscript[\[ScriptCapitalC], n - 1][\[Nu], a, b, z] + (((1 + a - n + \[Nu]) (1 + b - n + \[Nu]) (4 + a + b - 2 n + 2 \[Nu]))/ ((-2 + n - \[Nu]) (2 + a + b - n + \[Nu]) (2 + a + b - 2 n + 2 \[Nu]))) Subscript[\[ScriptCapitalC], n - 2][\[Nu], a, b, z] && Element[n, Integers] && n > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["JacobiP", "[", RowBox[List["\[Nu]", ",", "a", ",", "b", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "n"], "[", RowBox[List["\[Nu]", ",", "a", ",", "b", ",", "z"]], "]"]], " ", RowBox[List["JacobiP", "[", RowBox[List[RowBox[List["\[Nu]", "-", "n"]], ",", "a", ",", "b", ",", "z"]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "n", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["b", "-", "n", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "a", "+", "b", "-", RowBox[List["2", " ", "n"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "n", "-", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", "a", "+", "b", "-", "n", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["a", "+", "b", "-", RowBox[List["2", " ", "n"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]]], RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "1"]]], "[", RowBox[List["\[Nu]", ",", "a", ",", "b", ",", "z"]], "]"]], " ", RowBox[List["JacobiP", "[", RowBox[List[RowBox[List["\[Nu]", "-", "n", "-", "1"]], ",", "a", ",", "b", ",", "z"]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "0"], "[", RowBox[List["\[Nu]", ",", "a", ",", "b", ",", "z"]], "]"]], "\[Equal]", "1"]], "\[And]", RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "1"], "[", RowBox[List["\[Nu]", ",", "a", ",", "b", ",", "z"]], "]"]], "\[Equal]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]]]], ")"]]]], RowBox[List["2", " ", "\[Nu]", " ", RowBox[List["(", RowBox[List["a", "+", "b", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]]]]], " ", "\[And]", RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "n"], "[", RowBox[List["\[Nu]", ",", "a", ",", "b", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "a", "+", "b", "-", RowBox[List["2", " ", "n"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["a", "+", "b", "-", RowBox[List["2", " ", "n"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "a", "+", "b", "-", RowBox[List["2", " ", "n"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]]]], ")"]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "-", "n", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", "a", "+", "b", "-", "n", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["a", "+", "b", "-", RowBox[List["2", " ", "n"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]]], RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "1"]]], "[", RowBox[List["\[Nu]", ",", "a", ",", "b", ",", "z"]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "a", "-", "n", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", "b", "-", "n", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["4", "+", "a", "+", "b", "-", RowBox[List["2", " ", "n"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "n", "-", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "a", "+", "b", "-", "n", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "a", "+", "b", "-", RowBox[List["2", " ", "n"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]]], RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "2"]]], "[", RowBox[List["\[Nu]", ",", "a", ",", "b", ",", "z"]], "]"]]]]]]]], "\[And]", RowBox[List["Element", "[", RowBox[List["n", ",", "Integers"]], "]"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]

 MathML Form

 P ν ( a , b ) ( z ) ( a - n + ν ) ( b - n + ν ) ( a + b - 2 n + 2 ν + 2 ) ( n - ν - 1 ) ( a + b - n + ν + 1 ) ( a + b - 2 n + 2 ν ) 𝒞 n - 1 ( ν , a , b , z ) P ν - n - 1 ( a , b ) ( z ) + 𝒞 n ( ν , a , b , z ) P ν - n ( a , b ) ( z ) /; 𝒞 0 ( ν , a , b , z ) 1 𝒞 1 ( ν , a , b , z ) ( a + b + 2 ν - 1 ) ( a 2 - b 2 + z ( a + b + 2 ν - 2 ) ( a + b + 2 ν ) ) 2 ν ( a + b + ν ) ( a + b + 2 ν - 2 ) 𝒞 n ( ν , a , b , z ) ( a + b - 2 n + 2 ν + 1 ) ( a 2 - b 2 + z ( a + b - 2 n + 2 ν ) ( a + b - 2 n + 2 ν + 2 ) ) 2 ( - n + ν + 1 ) ( a + b - n + ν + 1 ) ( a + b - 2 n + 2 ν ) 𝒞 n - 1 ( ν , a , b , z ) + ( a - n + ν + 1 ) ( b - n + ν + 1 ) ( a + b - 2 n + 2 ν + 4 ) ( n - ν - 2 ) ( a + b - n + ν + 2 ) ( a + b - 2 n + 2 ν + 2 ) 𝒞 n - 2 ( ν , a , b , z ) n + Condition JacobiP ν a b z a -1 n ν b -1 n ν a b -1 2 n 2 ν 2 n -1 ν -1 a b -1 n ν 1 a b -1 2 n 2 ν -1 Subscript 𝒞 n -1 ν a b z JacobiP ν -1 n -1 a b z Subscript 𝒞 n ν a b z JacobiP ν -1 n a b z Subscript 𝒞 0 ν a b z 1 Subscript 𝒞 1 ν a b z a b 2 ν -1 a 2 -1 b 2 z a b 2 ν -2 a b 2 ν 2 ν a b ν a b 2 ν -2 -1 Subscript 𝒞 n ν a b z a b -1 2 n 2 ν 1 a 2 -1 b 2 z a b -1 2 n 2 ν a b -1 2 n 2 ν 2 2 -1 n ν 1 a b -1 n ν 1 a b -1 2 n 2 ν -1 Subscript 𝒞 n -1 ν a b z a -1 n ν 1 b -1 n ν 1 a b -1 2 n 2 ν 4 n -1 ν -2 a b -1 n ν 2 a b -1 2 n 2 ν 2 -1 Subscript 𝒞 n -2 ν a b z n SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["JacobiP", "[", RowBox[List["\[Nu]_", ",", "a_", ",", "b_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "n"], "[", RowBox[List["\[Nu]", ",", "a", ",", "b", ",", "z"]], "]"]], " ", RowBox[List["JacobiP", "[", RowBox[List[RowBox[List["\[Nu]", "-", "n"]], ",", "a", ",", "b", ",", "z"]], "]"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "n", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["b", "-", "n", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "a", "+", "b", "-", RowBox[List["2", " ", "n"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ")"]], " ", RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "1"]]], "[", RowBox[List["\[Nu]", ",", "a", ",", "b", ",", "z"]], "]"]], " ", RowBox[List["JacobiP", "[", RowBox[List[RowBox[List["\[Nu]", "-", "n", "-", "1"]], ",", "a", ",", "b", ",", "z"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "n", "-", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", "a", "+", "b", "-", "n", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["a", "+", "b", "-", RowBox[List["2", " ", "n"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "0"], "[", RowBox[List["\[Nu]", ",", "a", ",", "b", ",", "z"]], "]"]], "\[Equal]", "1"]], "&&", RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "1"], "[", RowBox[List["\[Nu]", ",", "a", ",", "b", ",", "z"]], "]"]], "\[Equal]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]]]], ")"]]]], RowBox[List["2", " ", "\[Nu]", " ", RowBox[List["(", RowBox[List["a", "+", "b", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]]]]], "&&", RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "n"], "[", RowBox[List["\[Nu]", ",", "a", ",", "b", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "a", "+", "b", "-", RowBox[List["2", " ", "n"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["a", "+", "b", "-", RowBox[List["2", " ", "n"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "a", "+", "b", "-", RowBox[List["2", " ", "n"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]]]], ")"]]]], ")"]], " ", RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "1"]]], "[", RowBox[List["\[Nu]", ",", "a", ",", "b", ",", "z"]], "]"]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "-", "n", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", "a", "+", "b", "-", "n", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["a", "+", "b", "-", RowBox[List["2", " ", "n"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "a", "-", "n", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", "b", "-", "n", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["4", "+", "a", "+", "b", "-", RowBox[List["2", " ", "n"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ")"]], " ", RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "2"]]], "[", RowBox[List["\[Nu]", ",", "a", ",", "b", ",", "z"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "n", "-", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "a", "+", "b", "-", "n", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "a", "+", "b", "-", RowBox[List["2", " ", "n"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]]]]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]

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

 2007-05-02