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variants of this functions
LegendreP






Mathematica Notation

Traditional Notation









Polynomials > LegendreP[n,z] > Series representations > Generalized power series > Expansions at generic point z==z0





http://functions.wolfram.com/05.03.06.0028.01









  


  










Input Form





LegendreP[n, z] == Sum[(2 k - 1)!! Sum[\[Ellipsis] Sum[KroneckerDelta[Sum[Subscript[i, j], {j, 1, 2 k + 1}], n - k] Product[LegendreP[Subscript[i, j], z] (z - Subscript[z, 0])^k, {j, 1, 2 k + 1}], {Subscript[i, 2 k + 1], 0, n - k}], {Subscript[i, 1], 0, n - k}], {k, 0, n}]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["n", ",", "z"]], "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], "-", "1"]], ")"]], "!!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["i", "1"], "=", "0"]], RowBox[List["n", "-", "k"]]], RowBox[List["\[Ellipsis]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["i", RowBox[List[RowBox[List["2", "k"]], "+", "1"]]], "=", "0"]], RowBox[List["n", "-", "k"]]], RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List[RowBox[List["2", "k"]], "+", "1"]]], SubscriptBox["i", "j"]]], ",", RowBox[List["n", "-", "k"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], RowBox[List[RowBox[List["2", "k"]], "+", "1"]]], RowBox[List[RowBox[List["LegendreP", "[", RowBox[List[SubscriptBox["i", "j"], ",", "z"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "k"]]]]]]]]]]]]]]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <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> &#63449; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mrow> <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> !! </mo> </mrow> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> i </mi> <mn> 1 </mn> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> </munderover> <mrow> <mo> &#8230; </mo> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> i </mi> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> </munderover> <mrow> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <msub> <mi> i </mi> <mi> j </mi> </msub> </mrow> <mo> , </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> </mrow> </msub> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mrow> <msub> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <msub> <mi> i </mi> <mi> j </mi> </msub> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> LegendreP </ci> <ci> n </ci> <ci> z </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <ci> Factorial2 </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> i </ci> <cn type='integer'> 1 </cn> </apply> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </uplimit> <apply> <times /> <ci> &#8230; </ci> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> i </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </uplimit> <apply> <times /> <apply> <ci> KroneckerDelta </ci> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> </uplimit> <apply> <ci> Subscript </ci> <ci> i </ci> <ci> j </ci> </apply> </apply> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> <apply> <product /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> </uplimit> <apply> <times /> <apply> <ci> LegendreP </ci> <apply> <ci> Subscript </ci> <ci> i </ci> <ci> j </ci> </apply> <ci> z </ci> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> </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_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "1"]], ")"]], "!!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["i", "1"], "=", "0"]], RowBox[List["n", "-", "k"]]], RowBox[List["\[Ellipsis]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["i", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]], "=", "0"]], RowBox[List["n", "-", "k"]]], RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]], SubscriptBox["i", "j"]]], ",", RowBox[List["n", "-", "k"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]], RowBox[List[RowBox[List["LegendreP", "[", RowBox[List[SubscriptBox["i", "j"], ",", "z"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "k"]]]]]]]]]]]]]]]]]]]]]










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





2007-05-02





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