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variants of this functions
GegenbauerC






Mathematica Notation

Traditional Notation









Hypergeometric Functions > GegenbauerC[nu,z] > Identities > Recurrence identities > Distant neighbors





http://functions.wolfram.com/07.13.17.0006.01









  


  










Input Form





GegenbauerC[\[Nu], z] == Subscript[\[ScriptCapitalC], m][\[Nu], z] GegenbauerC[\[Nu] - m, z] - ((\[Nu] - m - 1)/(\[Nu] - m + 1)) Subscript[\[ScriptCapitalC], m - 1][\[Nu], z] GegenbauerC[\[Nu] - m - 1, z] /; Subscript[\[ScriptCapitalC], 0][\[Nu], z] == 1 && Subscript[\[ScriptCapitalC], 1][\[Nu], z] == (2 (\[Nu] - 1) z)/\[Nu] && Subscript[\[ScriptCapitalC], m][\[Nu], z] == ((2 z (\[Nu] - m))/(\[Nu] - m + 1)) Subscript[\[ScriptCapitalC], m - 1][ \[Nu], z] - ((\[Nu] - m)/(\[Nu] - m + 2)) Subscript[\[ScriptCapitalC], m - 2][\[Nu], z] && Element[m, Integers] && m > 0










Standard Form





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MathML Form







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</ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> &#957; </ci> <ci> z </ci> </apply> <apply> <ci> GegenbauerC </ci> <apply> <plus /> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 0 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> &#119966; </ci> <cn type='integer'> 0 </cn> </apply> <ci> &#957; </ci> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> &#119966; </ci> <cn type='integer'> 1 </cn> </apply> <ci> &#957; </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> -1 </cn> </apply> <ci> z </ci> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> &#119966; </ci> <ci> m </ci> </apply> <ci> &#957; </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <apply> <plus /> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> &#957; </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> &#119966; </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> &#957; </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <plus /> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> &#957; </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> &#119966; </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -2 </cn> </apply> </apply> <ci> &#957; </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> m </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "m"], "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], " ", RowBox[List["GegenbauerC", "[", RowBox[List[RowBox[List["\[Nu]", "-", "m"]], ",", "z"]], "]"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", "-", "m", "-", "1"]], ")"]], " ", RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["m", "-", "1"]]], "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], " ", RowBox[List["GegenbauerC", "[", RowBox[List[RowBox[List["\[Nu]", "-", "m", "-", "1"]], ",", "z"]], "]"]]]], RowBox[List["\[Nu]", "-", "m", "+", "1"]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "0"], "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", "1"]], "&&", RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "1"], "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["\[Nu]", "-", "1"]], ")"]], " ", "z"]], "\[Nu]"]]], "&&", RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "m"], "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", "z", " ", RowBox[List["(", RowBox[List["\[Nu]", "-", "m"]], ")"]]]], ")"]], " ", RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["m", "-", "1"]]], "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], RowBox[List["\[Nu]", "-", "m", "+", "1"]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", "-", "m"]], ")"]], " ", RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["m", "-", "2"]]], "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], RowBox[List["\[Nu]", "-", "m", "+", "2"]]]]]]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]]]]]]]]]]










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





2007-05-02