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variants of this functions
LegendreP




Mathematica Notation

Traditional Notation









Polynomials > LegendreP[n,z] > Specific values > Specialized values > For fixed n





http://functions.wolfram.com/05.03.03.0018.01









  


  








Input Form





Derivative[0, r][LegendreP][n, -1] == ((-1)^(n + r) (n + r)!)/ (2^r r! (n - r)!) /; Element[n, Integers] && n >= 0 && Element[r, Integers] && r >= 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["LegendreP", TagBox[RowBox[List["(", RowBox[List["0", ",", "r"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["n", ",", RowBox[List["-", "1"]]]], "]"]], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "+", "r"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "+", "r"]], ")"]], "!"]]]], RowBox[List[SuperscriptBox["2", "r"], " ", RowBox[List["r", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "r"]], ")"]], "!"]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["r", "\[Element]", "Integers"]], "\[And]", RowBox[List["r", "\[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> <msubsup> <mi> P </mi> <mi> n </mi> <semantics> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mi> r </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;0&quot;, &quot;,&quot;, &quot;r&quot;]], &quot;)&quot;]], Derivative] </annotation> </semantics> </msubsup> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> + </mo> <mi> r </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mi> r </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> <mrow> <msup> <mn> 2 </mn> <mi> r </mi> </msup> <mo> &#8290; </mo> <mrow> <mi> r </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> r </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <apply> <partialdiff /> <list> <cn type='integer'> 0 </cn> <ci> r </ci> </list> <apply> <ci> Subscript </ci> <ci> P </ci> <ci> n </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> n </ci> <ci> r </ci> </apply> </apply> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <ci> r </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> r </ci> </apply> <apply> <factorial /> <ci> r </ci> </apply> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> r </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["LegendreP", TagBox[RowBox[List["(", RowBox[List["0", ",", "r"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["n_", ",", RowBox[List["-", "1"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "+", "r"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "+", "r"]], ")"]], "!"]]]], RowBox[List[SuperscriptBox["2", "r"], " ", RowBox[List["r", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "r"]], ")"]], "!"]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List["r", "\[Element]", "Integers"]], "&&", RowBox[List["r", "\[GreaterEqual]", "0"]]]]]]]]]]


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





2002-12-18