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 JacobiP

 http://functions.wolfram.com/07.15.17.0018.01

 Input Form

 JacobiP[\[Nu], a, b, z] == Subscript[\[ScriptCapitalC], n][\[Nu], a, b, z] JacobiP[\[Nu] + n, a, b, z] - (((n + 1 + \[Nu]) (n + 1 + a + b + \[Nu]) (2 n + a + b + 2 \[Nu]))/ ((n + a + \[Nu]) (n + b + \[Nu]) (2 n + 2 + a + b + 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] == ((3 + a + b + 2 \[Nu]) (a^2 - b^2 + z (2 + a + b + 2 \[Nu]) (4 + a + b + 2 \[Nu])))/(2 (1 + a + \[Nu]) (1 + b + \[Nu]) (4 + a + b + 2 \[Nu])) && Subscript[\[ScriptCapitalC], n][\[Nu], a, b, z] == (((2 n + a + b + 2 \[Nu] + 1) (a^2 - b^2 + z (2 n + a + b + 2 \[Nu]) (2 n + a + b + 2 \[Nu] + 2)))/ (2 (n + a + \[Nu]) (n + b + \[Nu]) (2 n + a + b + 2 \[Nu] + 2))) Subscript[\[ScriptCapitalC], n - 1][\[Nu], a, b, z] - (((n + \[Nu]) (n + a + b + \[Nu]) (2 n + a + b + 2 \[Nu] - 2))/ ((n + a + \[Nu] - 1) (n + b + \[Nu] - 1) (2 n + a + b + 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["n", "+", "1", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "+", "1", "+", "a", "+", "b", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", "n"]], "+", "a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "+", "a", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "+", "b", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", "n"]], "+", "2", "+", "a", "+", "b", "+", 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["3", "+", "a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["2", "+", "a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["4", "+", "a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]]]], ")"]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "a", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", "b", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["4", "+", "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[RowBox[List["2", "n"]], "+", "a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["2", "n"]], "+", "a", "+", "b", "+", RowBox[List["2", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", "n"]], "+", "a", "+", "b", "+", RowBox[List["2", "\[Nu]"]], "+", "2"]], ")"]]]]]], ")"]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["n", "+", "a", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "+", "b", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", "n"]], "+", "a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]], "+", "2"]], ")"]]]]], " ", RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "1"]]], "[", RowBox[List["\[Nu]", ",", "a", ",", "b", ",", "z"]], "]"]]]], "-", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["n", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "+", "a", "+", "b", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", "n"]], "+", "a", "+", "b", "+", RowBox[List["2", "\[Nu]"]], "-", "2"]], ")"]], " "]], RowBox[List[RowBox[List["(", RowBox[List["n", "+", "a", "+", "\[Nu]", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "+", "b", "+", "\[Nu]", "-", "1"]], ")"]], RowBox[List["(", RowBox[List[RowBox[List["2", "n"]], "+", "a", "+", "b", "+", 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 ) 𝒞 n ( ν , a , b , z ) P ν + n ( a , b ) ( z ) - ( n + ν + 1 ) ( a + b + n + ν + 1 ) ( a + b + 2 n + 2 ν ) ( a + n + ν ) ( b + n + ν ) ( a + b + 2 n + 2 ν + 2 ) 𝒞 n - 1 ( ν , a , b , z ) P ν + n + 1 ( a , b ) ( z ) /; 𝒞 0 ( ν , a , b , z ) 1 𝒞 1 ( ν , a , b , z ) ( a + b + 2 ν + 3 ) ( a 2 - b 2 + z ( a + b + 2 ν + 2 ) ( a + b + 2 ν + 4 ) ) 2 ( a + ν + 1 ) ( b + ν + 1 ) ( a + b + 2 ν + 4 ) 𝒞 n ( ν , a , b , z ) ( a + b + 2 n + 2 ν + 1 ) ( a 2 - b 2 + ( a + b + 2 n + 2 ν + 2 ) z ( a + b + 2 n + 2 ν ) ) 2 ( a + n + ν ) ( b + n + ν ) ( a + b + 2 n + 2 ν + 2 ) 𝒞 n - 1 ( ν , a , b , z ) - ( n + ν ) ( a + b + n + ν ) ( a + b + 2 n + 2 ν - 2 ) ( a + n + ν - 1 ) ( b + n + ν - 1 ) ( a + b + 2 n + 2 ν ) 𝒞 n - 2 ( ν , a , b , z ) n + Condition JacobiP ν a b z Subscript 𝒞 n ν a b z JacobiP ν n a b z -1 n ν 1 a b n ν 1 a b 2 n 2 ν a n ν b n ν a b 2 n 2 ν 2 -1 Subscript 𝒞 n -1 ν a b z JacobiP ν n 1 a b z Subscript 𝒞 0 ν a b z 1 Subscript 𝒞 1 ν a b z a b 2 ν 3 a 2 -1 b 2 z a b 2 ν 2 a b 2 ν 4 2 a ν 1 b ν 1 a b 2 ν 4 -1 Subscript 𝒞 n ν a b z a b 2 n 2 ν 1 a 2 -1 b 2 a b 2 n 2 ν 2 z a b 2 n 2 ν 2 a n ν b n ν a b 2 n 2 ν 2 -1 Subscript 𝒞 n -1 ν a b z -1 n ν a b n ν a b 2 n 2 ν -2 a n ν -1 b n ν -1 a b 2 n 2 ν -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["n", "+", "1", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "+", "1", "+", "a", "+", "b", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "a", "+", "b", "+", 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["n", "+", "a", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "+", "b", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "2", "+", "a", "+", "b", "+", 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["3", "+", "a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["2", "+", "a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["4", "+", "a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]]]], ")"]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "a", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", "b", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["4", "+", "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[RowBox[List["2", " ", "n"]], "+", "a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]], "+", "2"]], ")"]]]]]], ")"]]]], ")"]], " ", RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "1"]]], "[", RowBox[List["\[Nu]", ",", "a", ",", "b", ",", "z"]], "]"]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["n", "+", "a", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "+", "b", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]], "+", "2"]], ")"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["n", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "+", "a", "+", "b", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]], "-", "2"]], ")"]]]], ")"]], " ", RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "2"]]], "[", RowBox[List["\[Nu]", ",", "a", ",", "b", ",", "z"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "+", "a", "+", "\[Nu]", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "+", "b", "+", "\[Nu]", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "a", "+", "b", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]]]]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]

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

 2007-05-02