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http://functions.wolfram.com/05.07.17.0010.01
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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]])
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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]", "]"]]]]]]]]]]]]]
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<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["P", LegendreP] </annotation> </semantics> <mi> n </mi> <mi> m </mi> </msubsup> <mo> ( </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> β </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> a </mi> <mi> c </mi> </mfrac> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </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["(", GridBox[List[List[TagBox[RowBox[List["m", "+", "n"]], Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> ⁢ </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> ⁢ </mo> <mrow> <msubsup> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox["P", LegendreP] </annotation> </semantics> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> <mi> m </mi> </msubsup> <mo> ( </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> γ </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> <mo> ∧ </mo> <mrow> <mi> m </mi> <mo> ∈ </mo> <mi> ℤ </mi> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> ≤ </mo> <mi> m </mi> <mo> ≤ </mo> <mi> n </mi> </mrow> <mo> ∧ </mo> <mrow> <mn> 0 </mn> <mo> ≤ </mo> <mi> β </mi> <mo> ≤ </mo> <mi> π </mi> </mrow> <mo> ∧ </mo> <mrow> <mn> 0 </mn> <mo> ≤ </mo> <mi> γ </mi> <mo> ≤ </mo> <mi> π </mi> </mrow> <mo> ∧ </mo> <mrow> <mi> α </mi> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mi> β </mi> </mrow> <mo> - </mo> <mi> γ </mi> <mo> + </mo> <mi> π </mi> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> β </mi> <mo> + </mo> <mi> γ </mi> </mrow> <mo> ≤ </mo> <mi> π </mi> </mrow> <mo> ∧ </mo> <mrow> <mi> a </mi> <mo> ∈ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[Reals]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> a </mi> <mo> > </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> b </mi> <mo> ⩵ </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mfrac> <mrow> <mtext> </mtext> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> β </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> α </mi> <mo> ) </mo> </mrow> </mfrac> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> c </mi> <mo> ⩵ </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mfrac> <mrow> <mtext> </mtext> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> γ </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> α </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> β </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> γ </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <ci> ℕ </ci> </apply> <apply> <in /> <ci> m </ci> <ci> ℤ </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> β </ci> <pi /> </apply> <apply> <leq /> <cn type='integer'> 0 </cn> <ci> γ </ci> <pi /> </apply> <apply> <eq /> <ci> α </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> β </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> γ </ci> </apply> <pi /> </apply> </apply> <apply> <leq /> <apply> <plus /> <ci> β </ci> <ci> γ </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> β </ci> </apply> <apply> <power /> <apply> <sin /> <ci> α </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> γ </ci> </apply> <apply> <power /> <apply> <sin /> <ci> α </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| 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]", "]"]]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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| LegendreP[n,z] | | LegendreP[nu,z] | | LegendreP[nu,mu,z] | | LegendreP[nu,mu,2,z] | | LegendreP[nu,mu,3,z] | | |
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){kind=small}
