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variants of this functions
LegendreP






Mathematica Notation

Traditional Notation









Polynomials > LegendreP[n,mu,2,z] > Identities > Functional identities > Expansion with respect to parameters





http://functions.wolfram.com/05.07.17.0010.01









  


  










Input Form





LegendreP[n, m, Cos[\[Beta]]] == (-1)^m (a/c)^n Sum[Binomial[n + m, k] (-(b/a))^(n - k) LegendreP[n - k, m, Cos[\[Gamma]]], {k, 0, n - m}] /; Element[n, Integers] && n >= 0 && Element[m, Integers] && -n <= m <= n && 0 <= \[Beta] <= Pi && 0 <= \[Gamma] <= Pi && \[Alpha] == Pi - \[Beta] - \[Gamma] && \[Beta] + \[Gamma] <= Pi && Element[a, Reals] && a > 0 && b == a (Sin[\[Beta]]/Sin[\[Alpha]]) && c == a (Sin[\[Gamma]]/Sin[\[Alpha]])










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["n", ",", "m", ",", RowBox[List["Cos", "[", "\[Beta]", "]"]]]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", SuperscriptBox[RowBox[List["(", FractionBox["a", "c"], ")"]], "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "m"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "+", "m"]], ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["b", "a"]]], ")"]], RowBox[List["n", "-", "k"]]], " ", RowBox[List["LegendreP", "[", RowBox[List[RowBox[List["n", "-", "k"]], ",", "m", ",", RowBox[List["Cos", "[", "\[Gamma]", "]"]]]], "]"]]]]]]]]]], "/;", "\[IndentingNewLine]", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["-", "n"]], "\[LessEqual]", "m", "\[LessEqual]", "n"]], "\[And]", RowBox[List["0", "\[LessEqual]", "\[Beta]", "\[LessEqual]", "\[Pi]"]], "\[And]", RowBox[List["0", "\[LessEqual]", "\[Gamma]", "\[LessEqual]", "\[Pi]"]], "\[And]", RowBox[List["\[Alpha]", "\[Equal]", RowBox[List["\[Pi]", "-", "\[Beta]", "-", "\[Gamma]"]]]], "\[And]", RowBox[List[RowBox[List["\[Beta]", "+", "\[Gamma]"]], "\[LessEqual]", "\[Pi]"]], "\[And]", RowBox[List["a", "\[Element]", "Reals"]], "\[And]", RowBox[List["a", ">", "0"]], "\[And]", RowBox[List["b", "\[Equal]", RowBox[List["a", " ", FractionBox[RowBox[List["Sin", "[", "\[Beta]", "]"]], RowBox[List["Sin", "[", "\[Alpha]", "]"]]]]]]], "\[And]", RowBox[List["c", "\[Equal]", RowBox[List["a", " ", FractionBox[RowBox[List["Sin", "[", "\[Gamma]", "]"]], RowBox[List["Sin", "[", "\[Alpha]", "]"]]]]]]]]]]]]]










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> <msubsup> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mi> n </mi> <mi> m </mi> </msubsup> <mo> ( </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#946; </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> a </mi> <mi> c </mi> </mfrac> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> </mrow> </munderover> <mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mi> k </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[RowBox[List[&quot;m&quot;, &quot;+&quot;, &quot;n&quot;]], Identity, Rule[Editable, True]]], List[TagBox[&quot;k&quot;, Identity, Rule[Editable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mi> b </mi> <mi> a </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msubsup> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> <mi> m </mi> </msubsup> <mo> ( </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#947; </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <mi> &#8484; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> &#8804; </mo> <mi> m </mi> <mo> &#8804; </mo> <mi> n </mi> </mrow> <mo> &#8743; </mo> <mrow> <mn> 0 </mn> <mo> &#8804; </mo> <mi> &#946; </mi> <mo> &#8804; </mo> <mi> &#960; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mn> 0 </mn> <mo> &#8804; </mo> <mi> &#947; </mi> <mo> &#8804; </mo> <mi> &#960; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#945; </mi> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mi> &#946; </mi> </mrow> <mo> - </mo> <mi> &#947; </mi> <mo> + </mo> <mi> &#960; </mi> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> &#946; </mi> <mo> + </mo> <mi> &#947; </mi> </mrow> <mo> &#8804; </mo> <mi> &#960; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> a </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> a </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> b </mi> <mo> &#10869; </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mfrac> <mrow> <mtext> </mtext> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#946; </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#945; </mi> <mo> ) </mo> </mrow> </mfrac> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> c </mi> <mo> &#10869; </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mfrac> <mrow> <mtext> </mtext> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#947; </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#945; </mi> <mo> ) </mo> </mrow> </mfrac> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> LegendreP </ci> <ci> n </ci> <ci> m </ci> <apply> <cos /> <ci> &#946; </ci> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <apply> <power /> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> c </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> n </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </uplimit> <apply> <times /> <apply> <ci> Binomial </ci> <apply> <plus /> <ci> m </ci> <ci> n </ci> </apply> <ci> k </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> <apply> <ci> LegendreP </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <ci> m </ci> <apply> <cos /> <ci> &#947; </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <ci> &#8469; </ci> </apply> <apply> <in /> <ci> m </ci> <ci> &#8484; </ci> </apply> <apply> <leq /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <ci> m </ci> <ci> n </ci> </apply> <apply> <leq /> <cn type='integer'> 0 </cn> <ci> &#946; </ci> <pi /> </apply> <apply> <leq /> <cn type='integer'> 0 </cn> <ci> &#947; </ci> <pi /> </apply> <apply> <eq /> <ci> &#945; </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#946; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#947; </ci> </apply> <pi /> </apply> </apply> <apply> <leq /> <apply> <plus /> <ci> &#946; </ci> <ci> &#947; </ci> </apply> <pi /> </apply> <apply> <in /> <ci> a </ci> <reals /> </apply> <apply> <gt /> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <apply> <eq /> <ci> b </ci> <apply> <times /> <ci> a </ci> <apply> <times /> <apply> <sin /> <ci> &#946; </ci> </apply> <apply> <power /> <apply> <sin /> <ci> &#945; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <eq /> <ci> c </ci> <apply> <times /> <ci> a </ci> <apply> <times /> <apply> <sin /> <ci> &#947; </ci> </apply> <apply> <power /> <apply> <sin /> <ci> &#945; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreP", "[", RowBox[List["n_", ",", "m_", ",", RowBox[List["Cos", "[", "\[Beta]_", "]"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", SuperscriptBox[RowBox[List["(", FractionBox["a", "c"], ")"]], "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "m"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "+", "m"]], ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["b", "a"]]], ")"]], RowBox[List["n", "-", "k"]]], " ", RowBox[List["LegendreP", "[", RowBox[List[RowBox[List["n", "-", "k"]], ",", "m", ",", RowBox[List["Cos", "[", "\[Gamma]", "]"]]]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["-", "n"]], "\[LessEqual]", "m", "\[LessEqual]", "n"]], "&&", RowBox[List["0", "\[LessEqual]", "\[Beta]", "\[LessEqual]", "\[Pi]"]], "&&", RowBox[List["0", "\[LessEqual]", "\[Gamma]", "\[LessEqual]", "\[Pi]"]], "&&", RowBox[List["\[Alpha]", "\[Equal]", RowBox[List["\[Pi]", "-", "\[Beta]", "-", "\[Gamma]"]]]], "&&", RowBox[List[RowBox[List["\[Beta]", "+", "\[Gamma]"]], "\[LessEqual]", "\[Pi]"]], "&&", RowBox[List["a", "\[Element]", "Reals"]], "&&", RowBox[List["a", ">", "0"]], "&&", RowBox[List["b", "\[Equal]", FractionBox[RowBox[List["a", " ", RowBox[List["Sin", "[", "\[Beta]", "]"]]]], RowBox[List["Sin", "[", "\[Alpha]", "]"]]]]], "&&", RowBox[List["c", "\[Equal]", FractionBox[RowBox[List["a", " ", RowBox[List["Sin", "[", "\[Gamma]", "]"]]]], RowBox[List["Sin", "[", "\[Alpha]", "]"]]]]]]]]]]]]]










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





2002-12-18





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