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variants of this functions
LegendreP




Mathematica Notation

Traditional Notation









Hypergeometric Functions > LegendreP[nu,z] > Series representations > Other series representations





http://functions.wolfram.com/07.07.06.0028.01









  


  








Input Form





LegendreP[n, x y - Sqrt[1 - x^2] Sqrt[1 - y^2] Cos[\[Alpha]]] == LegendreP[n, x] LegendreP[n, y] + 2 Sum[((-1)^k Cos[k \[Alpha]] (n - k)! LegendreP[n, k, x] LegendreP[n, k, y])/(k + n)!, {k, 1, n}] /; Element[n, Integers] && n >= 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["n", ",", RowBox[List[RowBox[List["x", " ", "y"]], "-", RowBox[List[SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]], " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]], " ", RowBox[List["Cos", "[", "\[Alpha]", "]"]]]]]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["n", ",", "x"]], "]"]], " ", RowBox[List["LegendreP", "[", RowBox[List["n", ",", "y"]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Cos", "[", RowBox[List["k", " ", "\[Alpha]"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], "!"]], " ", RowBox[List["LegendreP", "[", RowBox[List["n", ",", "k", ",", "x"]], "]"]], " ", RowBox[List["LegendreP", "[", RowBox[List["n", ",", "k", ",", "y"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["k", "+", "n"]], ")"]], "!"]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "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> <msub> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mi> n </mi> </msub> <mo> ( </mo> <mrow> <mrow> <mi> x </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> <mo> - </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#945; </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mrow> <msub> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mi> n </mi> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mi> n </mi> </msub> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> &#8290; </mo> <mi> &#945; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msubsup> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mi> n </mi> <mi> k </mi> </msubsup> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msubsup> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mi> n </mi> <mi> k </mi> </msubsup> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> LegendreP </ci> <ci> n </ci> <apply> <plus /> <apply> <times /> <ci> x </ci> <ci> y </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <cos /> <ci> &#945; </ci> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> LegendreP </ci> <ci> n </ci> <ci> x </ci> </apply> <apply> <ci> LegendreP </ci> <ci> n </ci> <ci> y </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <cos /> <apply> <times /> <ci> k </ci> <ci> &#945; </ci> </apply> </apply> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <ci> k </ci> <ci> n </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> LegendreP </ci> <ci> n </ci> <ci> k </ci> <ci> x </ci> </apply> <apply> <ci> LegendreP </ci> <ci> n </ci> <ci> k </ci> <ci> y </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <ci> &#8469; </ci> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreP", "[", RowBox[List["n_", ",", RowBox[List[RowBox[List["x_", " ", "y_"]], "-", RowBox[List[SqrtBox[RowBox[List["1", "-", SuperscriptBox["x_", "2"]]]], " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y_", "2"]]]], " ", RowBox[List["Cos", "[", "\[Alpha]_", "]"]]]]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["n", ",", "x"]], "]"]], " ", RowBox[List["LegendreP", "[", RowBox[List["n", ",", "y"]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Cos", "[", RowBox[List["k", " ", "\[Alpha]"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], "!"]], " ", RowBox[List["LegendreP", "[", RowBox[List["n", ",", "k", ",", "x"]], "]"]], " ", RowBox[List["LegendreP", "[", RowBox[List["n", ",", "k", ",", "y"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["k", "+", "n"]], ")"]], "!"]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]


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





2001-10-29