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variants of this functions
LegendreP






Mathematica Notation

Traditional Notation









Hypergeometric Functions > LegendreP[nu,z] > Transformations > Products, sums, and powers of the direct function > Products of the direct function





http://functions.wolfram.com/07.07.16.0003.01









  


  










Input Form





LegendreP[n, z] LegendreP[m, z] == Sum[b[n, m, k] LegendreP[k, z], {k, Abs[m - n], m + n}] /; b[n, m, k] == KroneckerDelta[0, Mod[(1/2) (k + m + n), 1]] (1 + 2 k) (k + m - n - 1)!! (k - m + n - 1)!! (m + n - k - 1)!! ((k + m + n)!!/((k + m - n)!! (k - m + n)!! (m + n - k)!! (k + m + n + 1)!!)) && Element[n, Integers] && n >= 0 && Element[m, Integers] && m >= 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["n", ",", "z"]], "]"]], " ", RowBox[List["LegendreP", "[", RowBox[List["m", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["Abs", "[", RowBox[List["m", "-", "n"]], "]"]]]], RowBox[List["m", "+", "n"]]], RowBox[List[RowBox[List["b", "[", RowBox[List["n", ",", "m", ",", "k"]], "]"]], " ", RowBox[List["LegendreP", "[", RowBox[List["k", ",", "z"]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["b", "[", RowBox[List["n", ",", "m", ",", "k"]], "]"]], "\[Equal]", " ", RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List["0", ",", RowBox[List["Mod", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["k", "+", "m", "+", "n"]], ")"]]]], ",", "1"]], "]"]]]], "]"]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "m", "-", "n", "-", "1"]], ")"]], "!!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "m", "+", "n", "-", "1"]], ")"]], "!!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["m", "+", "n", "-", "k", "-", "1"]], ")"]], "!!"]], " ", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "+", "m", "+", "n"]], ")"]], "!!"]], "/", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "+", "m", "-", "n"]], ")"]], "!!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "m", "+", "n"]], ")"]], "!!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["m", "+", "n", "-", "k"]], ")"]], "!!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "m", "+", "n", "+", "1"]], ")"]], "!!"]]]], ")"]]]]]]]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[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> <mrow> <msub> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mi> n </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mi> m </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> </mrow> </munderover> <mrow> <mrow> <mi> b </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> m </mi> <mo> , </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mi> k </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> b </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> m </mi> <mo> , </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mn> 0 </mn> <mo> , </mo> <semantics> <mrow> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> m </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mn> 1 </mn> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> $CellContext`k </ci> <ci> FE`Conversion`Private`m </ci> <ci> FE`Conversion`Private`n </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </annotation-xml> </semantics> </mrow> </msub> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> m </mi> <mo> - </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> !! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> m </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> !! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> !! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> m </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> !! </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> m </mi> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> !! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> m </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> !! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mo> !! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> m </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> !! </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <apply> <ci> LegendreP </ci> <ci> n </ci> <ci> z </ci> </apply> <apply> <ci> LegendreP </ci> <ci> m </ci> <ci> z </ci> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <apply> <abs /> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> </lowlimit> <uplimit> <apply> <plus /> <ci> m </ci> <ci> n </ci> </apply> </uplimit> <apply> <times /> <apply> <ci> b </ci> <ci> n </ci> <ci> m </ci> <ci> k </ci> </apply> <apply> <ci> LegendreP </ci> <ci> k </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <ci> b </ci> <ci> n </ci> <ci> m </ci> <ci> k </ci> </apply> <apply> <times /> <apply> <times /> <apply> <ci> KroneckerDelta </ci> <cn type='integer'> 0 </cn> <apply> <rem /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> $CellContext`k </ci> <ci> FE`Conversion`Private`m </ci> <ci> FE`Conversion`Private`n </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Factorial2 </ci> <apply> <plus /> <ci> k </ci> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Factorial2 </ci> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Factorial2 </ci> <apply> <plus /> <ci> m </ci> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Factorial2 </ci> <apply> <plus /> <ci> k </ci> <ci> m </ci> <ci> n </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Factorial2 </ci> <apply> <plus /> <ci> k </ci> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <ci> Factorial2 </ci> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <ci> n </ci> </apply> </apply> <apply> <ci> Factorial2 </ci> <apply> <plus /> <ci> m </ci> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> <apply> <ci> Factorial2 </ci> <apply> <plus /> <ci> k </ci> <ci> m </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <ci> &#8469; </ci> </apply> <apply> <in /> <ci> m </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["n_", ",", "z_"]], "]"]], " ", RowBox[List["LegendreP", "[", RowBox[List["m_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["Abs", "[", RowBox[List["m", "-", "n"]], "]"]]]], RowBox[List["m", "+", "n"]]], RowBox[List[RowBox[List["b", "[", RowBox[List["n", ",", "m", ",", "k"]], "]"]], " ", RowBox[List["LegendreP", "[", RowBox[List["k", ",", "z"]], "]"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["b", "[", RowBox[List["n", ",", "m", ",", "k"]], "]"]], "\[Equal]", FractionBox[RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List["0", ",", RowBox[List["Mod", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["k", "+", "m", "+", "n"]], ")"]]]], ",", "1"]], "]"]]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "m", "-", "n", "-", "1"]], ")"]], "!!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "m", "+", "n", "-", "1"]], ")"]], "!!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["m", "+", "n", "-", "k", "-", "1"]], ")"]], "!!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "m", "+", "n"]], ")"]], "!!"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "+", "m", "-", "n"]], ")"]], "!!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "m", "+", "n"]], ")"]], "!!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["m", "+", "n", "-", "k"]], ")"]], "!!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "m", "+", "n", "+", "1"]], ")"]], "!!"]]]]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]]]










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





2001-10-29





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