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variants of this functions
GegenbauerC






Mathematica Notation

Traditional Notation









Hypergeometric Functions > GegenbauerC[nu,lambda,z] > Differentiation > Low-order differentiation > With respect to z





http://functions.wolfram.com/07.14.20.0006.01









  


  










Input Form





D[(1 - z^2)^(\[Lambda] - 1/2) GegenbauerC[\[Nu], \[Lambda], z], z] == (-(((\[Nu] + 1) (2 \[Lambda] + \[Nu] - 1))/(2 (\[Lambda] - 1)))) (1 - z^2)^(\[Lambda] - 3/2) GegenbauerC[\[Nu] + 1, \[Lambda] - 1, z]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", "z"], RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["\[Lambda]", "-", FractionBox["1", "2"]]]], RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]", ",", "\[Lambda]", ",", "z"]], "]"]]]], ")"]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "\[Nu]", "-", "1"]], " ", ")"]]]], RowBox[List["2", RowBox[List["(", RowBox[List["\[Lambda]", "-", "1"]], ")"]]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["\[Lambda]", "-", FractionBox["3", "2"]]]], RowBox[List["GegenbauerC", "[", RowBox[List[RowBox[List["\[Nu]", "+", "1"]], ",", RowBox[List["\[Lambda]", "-", "1"]], ",", "z"]], "]"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mfrac> <mrow> <mo> &#8706; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> &#955; </mi> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msubsup> <mi> C </mi> <mi> &#957; </mi> <mi> &#955; </mi> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <mi> z </mi> </mrow> </mfrac> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#955; </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#955; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> &#955; </mi> <mo> - </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msubsup> <mi> C </mi> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mi> &#955; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <ci> &#955; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> C </ci> <ci> &#957; </ci> </apply> <ci> &#955; </ci> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#955; </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> &#955; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <ci> &#955; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> C </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <plus /> <ci> &#955; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> z </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["z_"]]], RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["z_", "2"]]], ")"]], RowBox[List["\[Lambda]_", "-", FractionBox["1", "2"]]]], " ", RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]_", ",", "\[Lambda]_", ",", "z_"]], "]"]]]], ")"]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "\[Nu]", "-", "1"]], ")"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["\[Lambda]", "-", FractionBox["3", "2"]]]], " ", RowBox[List["GegenbauerC", "[", RowBox[List[RowBox[List["\[Nu]", "+", "1"]], ",", RowBox[List["\[Lambda]", "-", "1"]], ",", "z"]], "]"]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["\[Lambda]", "-", "1"]], ")"]]]]]]]]]]]










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





2001-10-29