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variants of this functions
LegendreP






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/07.09.17.0008.01









  


  










Input Form





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










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", RowBox[List["\[Mu]", "+", "1"]], ",", "3", ",", "z"]], "]"]], RowBox[List["LegendreP", "[", RowBox[List[SubscriptBox["\[Nu]", "1"], ",", RowBox[List[SubscriptBox["\[Mu]", "1"], "+", "1"]], ",", "3", ",", "z"]], "]"]]]], "-", RowBox[List[RowBox[List["LegendreP", "[", RowBox[List[RowBox[List["\[Nu]", "-", "1"]], ",", RowBox[List["\[Mu]", "+", "1"]], ",", "3", ",", "z"]], "]"]], RowBox[List["LegendreP", "[", RowBox[List[RowBox[List[SubscriptBox["\[Nu]", "1"], "-", "1"]], ",", RowBox[List[SubscriptBox["\[Mu]", "1"], "+", "1"]], ",", "3", ",", "z"]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", "-", "\[Mu]"]], ")"]], RowBox[List["(", RowBox[List[SubscriptBox["\[Nu]", "1"], "-", SubscriptBox["\[Mu]", "1"]]], ")"]], " ", RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "3", ",", "z"]], "]"]], RowBox[List["LegendreP", "[", RowBox[List[SubscriptBox["\[Nu]", "1"], ",", SubscriptBox["\[Mu]", "1"], ",", "3", ",", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["\[Mu]", "+", "\[Nu]"]], ")"]], RowBox[List["(", RowBox[List[SubscriptBox["\[Mu]", "1"], "+", SubscriptBox["\[Nu]", "1"]]], ")"]], " ", RowBox[List["LegendreP", "[", RowBox[List[RowBox[List["\[Nu]", "-", "1"]], ",", "\[Mu]", ",", "3", ",", "z"]], "]"]], RowBox[List["LegendreP", "[", RowBox[List[RowBox[List[SubscriptBox["\[Nu]", "1"], "-", "1"]], ",", SubscriptBox["\[Mu]", "1"], ",", "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> <mrow> <msubsup> <semantics> <mi> &#120083; </mi> <annotation encoding='Mathematica'> TagBox[TagBox[&quot;\[GothicCapitalP]&quot;, LegendreQ], 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, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msubsup> <semantics> <mi> &#120083; </mi> <annotation encoding='Mathematica'> TagBox[TagBox[&quot;\[GothicCapitalP]&quot;, LegendreQ], LegendreP] </annotation> </semantics> <msub> <mi> &#957; </mi> <mn> 1 </mn> </msub> <mrow> <msub> <mi> &#956; </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> ( </mo> <semantics> <mi> z </mi> <annotation encoding='Mathematica'> TagBox[&quot;z&quot;, HoldComplete[LegendreP, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <msubsup> <semantics> <mi> &#120083; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[GothicCapitalP]&quot;, LegendreQ] </annotation> </semantics> <mrow> <mi> &#957; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <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, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msubsup> <semantics> <mi> &#120083; </mi> <annotation encoding='Mathematica'> TagBox[TagBox[&quot;\[GothicCapitalP]&quot;, LegendreQ], LegendreP] </annotation> </semantics> <mrow> <msub> <mi> &#957; </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <msub> <mi> &#956; </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> ( </mo> <semantics> <mi> z </mi> <annotation encoding='Mathematica'> TagBox[&quot;z&quot;, HoldComplete[LegendreP, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> &#957; 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</mi> <mo> + </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> &#956; </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> &#957; </mi> <mn> 1 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msubsup> <semantics> <mi> &#120083; </mi> <annotation encoding='Mathematica'> TagBox[TagBox[&quot;\[GothicCapitalP]&quot;, LegendreQ], 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, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msubsup> <semantics> <mi> &#120083; </mi> <annotation encoding='Mathematica'> TagBox[TagBox[&quot;\[GothicCapitalP]&quot;, LegendreQ], LegendreP] </annotation> </semantics> <mrow> <msub> <mi> &#957; </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> <msub> <mi> &#956; </mi> <mn> 1 </mn> </msub> </msubsup> <mo> ( </mo> <semantics> <mi> z </mi> <annotation encoding='Mathematica'> TagBox[&quot;z&quot;, HoldComplete[LegendreP, 2]] </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 /> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <ci> LegendreP </ci> <apply> <ci> LegendreQ </ci> <ci> &#120083; </ci> </apply> </apply> <ci> &#957; </ci> </apply> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <apply> <ci> HoldComplete </ci> <ci> LegendreP </ci> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <ci> LegendreP </ci> <apply> <ci> LegendreQ </ci> <ci> &#120083; </ci> </apply> </apply> <apply> <ci> Subscript </ci> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> &#956; </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <apply> <ci> HoldComplete </ci> <ci> LegendreP </ci> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <ci> LegendreQ </ci> <ci> &#120083; </ci> </apply> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <apply> <ci> HoldComplete </ci> <ci> LegendreP </ci> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <ci> LegendreP </ci> <apply> <ci> LegendreQ </ci> <ci> &#120083; </ci> </apply> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> &#956; </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <apply> <ci> HoldComplete </ci> <ci> LegendreP </ci> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> &#956; </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <ci> LegendreP </ci> <apply> <ci> LegendreQ </ci> <ci> &#120083; </ci> </apply> </apply> <ci> &#957; </ci> </apply> <ci> &#956; </ci> </apply> <apply> <apply> <ci> HoldComplete </ci> <ci> LegendreP </ci> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <ci> LegendreP </ci> <apply> <ci> LegendreQ </ci> <ci> &#120083; </ci> </apply> </apply> <apply> <ci> Subscript </ci> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> &#956; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <apply> <ci> HoldComplete </ci> <ci> LegendreP </ci> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> &#956; </ci> <ci> &#957; </ci> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> &#956; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <ci> LegendreP </ci> <apply> <ci> LegendreQ </ci> <ci> &#120083; </ci> </apply> </apply> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> &#956; </ci> </apply> <apply> <apply> <ci> HoldComplete </ci> <ci> LegendreP </ci> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <ci> LegendreP </ci> <apply> <ci> LegendreQ </ci> <ci> &#120083; </ci> </apply> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> &#956; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <apply> <ci> HoldComplete </ci> <ci> LegendreP </ci> <cn type='integer'> 2 </cn> </apply> <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[RowBox[List["LegendreP", "[", RowBox[List["\[Nu]_", ",", RowBox[List["\[Mu]_", "+", "1"]], ",", "3", ",", "z_"]], "]"]], " ", RowBox[List["LegendreP", "[", RowBox[List[SubscriptBox["\[Nu]_", "1"], ",", RowBox[List[SubscriptBox["\[Mu]_", "1"], "+", "1"]], ",", "3", ",", "z_"]], "]"]]]], "-", RowBox[List[RowBox[List["LegendreP", "[", RowBox[List[RowBox[List["\[Nu]_", "-", "1"]], ",", RowBox[List["\[Mu]_", "+", "1"]], ",", "3", ",", "z_"]], "]"]], " ", RowBox[List["LegendreP", "[", RowBox[List[RowBox[List[SubscriptBox["\[Nu]_", "1"], "-", "1"]], ",", RowBox[List[SubscriptBox["\[Mu]_", "1"], "+", "1"]], ",", "3", ",", "z_"]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["\[Nu]_", "-", "\[Mu]_"]], ")"]], " ", RowBox[List["(", RowBox[List[SubscriptBox["\[Nu]_", "1"], "-", SubscriptBox["\[Mu]_", "1"]]], ")"]], " ", RowBox[List["LegendreP", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "3", ",", "z_"]], "]"]], " ", RowBox[List["LegendreP", "[", RowBox[List[SubscriptBox["\[Nu]_", "1"], ",", SubscriptBox["\[Mu]_", "1"], ",", "3", ",", "z_"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["\[Mu]_", "+", "\[Nu]_"]], ")"]], " ", RowBox[List["(", RowBox[List[SubscriptBox["\[Mu]_", "1"], "+", SubscriptBox["\[Nu]_", "1"]]], ")"]], " ", RowBox[List["LegendreP", "[", RowBox[List[RowBox[List["\[Nu]_", "-", "1"]], ",", "\[Mu]_", ",", "3", ",", "z_"]], "]"]], " ", RowBox[List["LegendreP", "[", RowBox[List[RowBox[List[SubscriptBox["\[Nu]_", "1"], "-", "1"]], ",", SubscriptBox["\[Mu]_", "1"], ",", "3", ",", "z_"]], "]"]]]]]], "]"]], "\[RuleDelayed]", "0"]]]]










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





2001-10-29





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