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WhittakerW






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/07.45.17.0005.01









  


  










Input Form





WhittakerW[\[Nu], \[Mu], z] == Subscript[\[ScriptCapitalC], n][\[Nu], \[Mu], z] WhittakerW[\[Nu] + n, \[Mu], z] + (4/((1 - 2 n + 2 \[Mu] - 2 \[Nu]) (-1 + 2 n + 2 \[Mu] + 2 \[Nu]))) Subscript[\[ScriptCapitalC], n - 1][\[Nu], \[Mu], z] WhittakerW[\[Nu] + n + 1, \[Mu], z] /; Subscript[\[ScriptCapitalC], 0][\[Nu], \[Mu], z] == 1 && Subscript[\[ScriptCapitalC], 1][\[Nu], \[Mu], z] == (4 (z - 2 (1 + \[Nu])))/(-4 \[Mu]^2 + (1 + 2 \[Nu])^2) && Subscript[\[ScriptCapitalC], n][\[Nu], \[Mu], z] == ((4 (z - 2 (1 + \[Nu])))/(-4 \[Mu]^2 + (1 + 2 \[Nu])^2)) Subscript[\[ScriptCapitalC], n - 1][\[Nu], \[Mu], z] - (2/((-(3/2) + n - \[Mu] + \[Nu]) (-3 + 2 n + 2 \[Mu] + 2 \[Nu]))) Subscript[\[ScriptCapitalC], n - 2][\[Nu], \[Mu], z] && Element[n, Integers] && n > 0










Standard Form





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MathML Form







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</mi> <mo> , </mo> <mi> &#956; </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mtext> </mtext> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> &#956; </mi> <mo> + </mo> <mi> &#957; </mi> <mo> - </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <mi> &#119966; </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </msub> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> , </mo> <mi> &#956; </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8712; 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</ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <apply> <ci> Subscript </ci> <ci> &#119966; </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> &#957; </ci> <ci> &#956; </ci> <ci> z </ci> </apply> <apply> <ci> WhittakerW </ci> <apply> <plus /> <ci> &#957; </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <ci> &#956; </ci> <ci> z </ci> </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> &#956; </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> &#956; </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> &#119966; </ci> <ci> n </ci> </apply> <ci> &#957; </ci> <ci> &#956; </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#957; 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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["WhittakerW", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "n"], "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]], " ", RowBox[List["WhittakerW", "[", RowBox[List[RowBox[List["\[Nu]", "+", "n"]], ",", "\[Mu]", ",", "z"]], "]"]]]], "+", FractionBox[RowBox[List["4", " ", RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "1"]]], "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]], " ", RowBox[List["WhittakerW", "[", RowBox[List[RowBox[List["\[Nu]", "+", "n", "+", "1"]], ",", "\[Mu]", ",", "z"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "n"]], "+", RowBox[List["2", " ", "\[Mu]"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]], "+", RowBox[List["2", " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "0"], "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]], "\[Equal]", "1"]], "&&", RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "1"], "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]], "\[Equal]", FractionBox[RowBox[List["4", " ", RowBox[List["(", RowBox[List["z", "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]]]]]], ")"]]]], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["\[Mu]", "2"]]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], "2"]]]]]], "&&", RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "n"], "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["4", " ", RowBox[List["(", RowBox[List["z", "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]]]]]], ")"]]]], ")"]], " ", RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "1"]]], "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox["\[Mu]", "2"]]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], "2"]]]], "-", FractionBox[RowBox[List["2", " ", RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "2"]]], "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["3", "2"]]], "+", "n", "-", "\[Mu]", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["2", " ", "n"]], "+", RowBox[List["2", " ", "\[Mu]"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]]]]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]










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





2007-05-02





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