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variants of this functions
LegendreP




Mathematica Notation

Traditional Notation









Hypergeometric Functions > LegendreP[nu,mu,3,z] > Identities > Functional identities > Relations between contiguous functions





http://functions.wolfram.com/07.09.17.0013.01









  


  








Input Form





z (1 + \[Nu] + \[Mu]) LegendreP[\[Nu], \[Mu], 3, z] + Sqrt[z - 1] Sqrt[1 + z] LegendreP[\[Nu], 1 + \[Mu], 3, z] - (1 + \[Nu] - \[Mu]) LegendreP[1 + \[Nu], \[Mu], 3, z] == 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["z", " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]", "+", "\[Mu]"]], ")"]], " ", RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "3", ",", "z"]], "]"]]]], "+", RowBox[List[SqrtBox[RowBox[List["z", "-", "1"]]], " ", SqrtBox[RowBox[List["1", "+", "z"]]], " ", RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", RowBox[List["1", "+", "\[Mu]"]], ",", "3", ",", "z"]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[Nu]", "-", "\[Mu]"]], ")"]], " ", RowBox[List["LegendreP", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "\[Mu]", ",", "3", ",", "z"]], "]"]]]]]], "\[Equal]", "0"]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> + </mo> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msubsup> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mi> &#957; </mi> <mi> &#956; </mi> </msubsup> <mo> ( </mo> <semantics> <mi> z </mi> <annotation encoding='Mathematica'> TagBox[&quot;z&quot;, HoldComplete[LegendreP, 3]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msqrt> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <msubsup> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mi> &#957; </mi> <mrow> <mi> &#956; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> ( </mo> <semantics> <mi> z </mi> <annotation encoding='Mathematica'> TagBox[&quot;z&quot;, HoldComplete[LegendreP, 3]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#956; </mi> </mrow> <mo> + </mo> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msubsup> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mi> &#956; </mi> </msubsup> <mo> ( </mo> <semantics> <mi> z </mi> <annotation encoding='Mathematica'> TagBox[&quot;z&quot;, HoldComplete[LegendreP, 3]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <plus /> <ci> &#956; </ci> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> LegendreP </ci> <ci> &#957; </ci> <ci> &#956; </ci> <cn type='integer'> 3 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> LegendreP </ci> <ci> &#957; </ci> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 3 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> LegendreP </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> <ci> &#956; </ci> <cn type='integer'> 3 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["z_", " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]_", "+", "\[Mu]_"]], ")"]], " ", RowBox[List["LegendreP", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "3", ",", "z_"]], "]"]]]], "+", RowBox[List[SqrtBox[RowBox[List["z_", "-", "1"]]], " ", SqrtBox[RowBox[List["1", "+", "z_"]]], " ", RowBox[List["LegendreP", "[", RowBox[List["\[Nu]_", ",", RowBox[List["1", "+", "\[Mu]_"]], ",", "3", ",", "z_"]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[Nu]_", "-", "\[Mu]_"]], ")"]], " ", RowBox[List["LegendreP", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]_"]], ",", "\[Mu]_", ",", "3", ",", "z_"]], "]"]]]]]], "]"]], "\[RuleDelayed]", "0"]]]]










Contributed by





Pavlyk O. (2006)


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





2007-05-02