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http://functions.wolfram.com/05.09.17.0010.01
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GegenbauerC[n, \[Lambda], z] ==
Subscript[\[ScriptCapitalC], m][n, \[Lambda], z]
GegenbauerC[n - m, \[Lambda], z] -
((2 \[Lambda] + n - m - 1)/(1 - m + n))
Subscript[\[ScriptCapitalC], m - 1][n, \[Lambda], z]
GegenbauerC[n - m - 1, \[Lambda], z] /;
Subscript[\[ScriptCapitalC], 0][n, \[Lambda], z] == 1 &&
Subscript[\[ScriptCapitalC], 1][n, \[Lambda], z] ==
(2 (\[Lambda] + n - 1) z)/n && Subscript[\[ScriptCapitalC], m][n,
\[Lambda], z] == ((2 z (\[Lambda] + n - m))/(1 - m + n))
Subscript[\[ScriptCapitalC], m - 1][n, \[Lambda], z] -
((2 \[Lambda] + n - m)/(2 - m + n)) Subscript[\[ScriptCapitalC], m - 2][
n, \[Lambda], z] && Element[m, Integers] && m > 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["GegenbauerC", "[", RowBox[List["n", ",", "\[Lambda]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "m"], "[", RowBox[List["n", ",", "\[Lambda]", ",", "z"]], "]"]], " ", RowBox[List["GegenbauerC", "[", RowBox[List[RowBox[List["n", "-", "m"]], ",", "\[Lambda]", ",", "z"]], "]"]]]], "-", RowBox[List[FractionBox[RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "n", "-", "m", "-", "1"]], RowBox[List["1", "-", "m", "+", "n"]]], " ", RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["m", "-", "1"]]], "[", RowBox[List["n", ",", "\[Lambda]", ",", "z"]], "]"]], " ", RowBox[List["GegenbauerC", "[", RowBox[List[RowBox[List["n", "-", "m", "-", "1"]], ",", "\[Lambda]", ",", "z"]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "0"], "[", RowBox[List["n", ",", "\[Lambda]", ",", "z"]], "]"]], "\[Equal]", "1"]], "\[And]", RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "1"], "[", RowBox[List["n", ",", "\[Lambda]", ",", "z"]], "]"]], "\[Equal]", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["\[Lambda]", "+", "n", "-", "1"]], ")"]], " ", "z"]], "n"]]], " ", "\[And]", RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "m"], "[", RowBox[List["n", ",", "\[Lambda]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List["2", " ", "z", " ", RowBox[List["(", RowBox[List["\[Lambda]", "+", "n", "-", "m"]], ")"]], " "]], RowBox[List["1", "-", "m", "+", "n"]]], RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["m", "-", "1"]]], "[", RowBox[List["n", ",", "\[Lambda]", ",", "z"]], "]"]]]], "-", RowBox[List[FractionBox[RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "n", "-", "m", " "]], RowBox[List["2", "-", "m", "+", "n"]]], RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["m", "-", "2"]]], "[", RowBox[List["n", ",", "\[Lambda]", ",", "z"]], "]"]]]]]]]], "\[And]", RowBox[List["Element", "[", RowBox[List["m", ",", "Integers"]], "]"]], "\[And]", RowBox[List["m", ">", "0"]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msubsup> <mi> C </mi> <mi> n </mi> <mi> λ </mi> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mrow> <msub> <mi> 𝒞 </mi> <mi> m </mi> </msub> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> λ </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msubsup> <mi> C </mi> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> </mrow> <mi> λ </mi> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <msub> <mi> 𝒞 </mi> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> λ </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msubsup> <mi> C </mi> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> λ </mi> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <msub> <mi> 𝒞 </mi> <mn> 0 </mn> </msub> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> λ </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <msub> <mi> 𝒞 </mi> <mn> 1 </mn> </msub> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> λ </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mi> λ </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mi> n </mi> </mfrac> </mrow> <mo> ∧ </mo> <mrow> <mrow> <msub> <mi> 𝒞 </mi> <mi> m </mi> </msub> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> λ </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> <mo> + </mo> <mi> λ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <msub> <mi> 𝒞 </mi> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> λ </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <msub> <mi> 𝒞 </mi> <mrow> <mi> m </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </msub> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> λ </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> C </ci> <ci> n </ci> </apply> <ci> λ </ci> </apply> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> 𝒞 </ci> <ci> m </ci> </apply> <ci> n </ci> <ci> λ </ci> <ci> z </ci> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> C </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <ci> λ </ci> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> λ </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <apply> <ci> Subscript </ci> <ci> 𝒞 </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> n </ci> <ci> λ </ci> <ci> z </ci> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> C </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> λ </ci> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> 𝒞 </ci> <cn type='integer'> 0 </cn> </apply> <ci> n </ci> <ci> λ </ci> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> 𝒞 </ci> <cn type='integer'> 1 </cn> </apply> <ci> n </ci> <ci> λ </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> n </ci> <ci> λ </ci> <cn type='integer'> -1 </cn> </apply> <ci> z </ci> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> 𝒞 </ci> <ci> m </ci> </apply> <ci> n </ci> <ci> λ </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <ci> λ </ci> </apply> <apply> <power /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <apply> <ci> Subscript </ci> <ci> 𝒞 </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> n </ci> <ci> λ </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> λ </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <apply> <ci> Subscript </ci> <ci> 𝒞 </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -2 </cn> </apply> </apply> <ci> n </ci> <ci> λ </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["GegenbauerC", "[", RowBox[List["n_", ",", "\[Lambda]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "m"], "[", RowBox[List["n", ",", "\[Lambda]", ",", "z"]], "]"]], " ", RowBox[List["GegenbauerC", "[", RowBox[List[RowBox[List["n", "-", "m"]], ",", "\[Lambda]", ",", "z"]], "]"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "n", "-", "m", "-", "1"]], ")"]], " ", RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["m", "-", "1"]]], "[", RowBox[List["n", ",", "\[Lambda]", ",", "z"]], "]"]], " ", RowBox[List["GegenbauerC", "[", RowBox[List[RowBox[List["n", "-", "m", "-", "1"]], ",", "\[Lambda]", ",", "z"]], "]"]]]], RowBox[List["1", "-", "m", "+", "n"]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "0"], "[", RowBox[List["n", ",", "\[Lambda]", ",", "z"]], "]"]], "\[Equal]", "1"]], "&&", RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "1"], "[", RowBox[List["n", ",", "\[Lambda]", ",", "z"]], "]"]], "\[Equal]", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["\[Lambda]", "+", "n", "-", "1"]], ")"]], " ", "z"]], "n"]]], "&&", RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "m"], "[", RowBox[List["n", ",", "\[Lambda]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", "z", " ", RowBox[List["(", RowBox[List["\[Lambda]", "+", "n", "-", "m"]], ")"]]]], ")"]], " ", RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["m", "-", "1"]]], "[", RowBox[List["n", ",", "\[Lambda]", ",", "z"]], "]"]]]], RowBox[List["1", "-", "m", "+", "n"]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "n", "-", "m"]], ")"]], " ", RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["m", "-", "2"]]], "[", RowBox[List["n", ",", "\[Lambda]", ",", "z"]], "]"]]]], RowBox[List["2", "-", "m", "+", "n"]]]]]]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
