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variants of this functions
LegendreP






Mathematica Notation

Traditional Notation









Hypergeometric Functions > LegendreP[nu,mu,3,z] > Series representations > Generalized power series > Expansions at z==0





http://functions.wolfram.com/07.09.06.0006.01









  


  










Input Form





LegendreP[n, m, 3, z] \[Proportional] ((-1)^(m + Floor[(n - m)/2]) ((z + 1)^(m/2)/(z - 1)^(m/2)) Binomial[n, Floor[(n - m)/2]] Binomial[2 n - 2 Floor[(n - m)/2], n] Pochhammer[1 - m + n - 2 Floor[(n - m)/2], m] z^(n - m - 2 Floor[(n - m)/2]) (1 + O[z]))/2^n /; (z -> 0) && Element[m, Integers] && m >= 0 && Element[n, Integers] && n >= 0 && n >= m










Standard Form





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MathML Form







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</mo> <mfrac> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8971; </mo> </mrow> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;n&quot;, &quot;-&quot;, &quot;m&quot;, &quot;-&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, RowBox[List[&quot;\[LeftFloor]&quot;, FractionBox[RowBox[List[&quot;n&quot;, &quot;-&quot;, &quot;m&quot;]], &quot;2&quot;], &quot;\[RightFloor]&quot;]]]], &quot;+&quot;, &quot;1&quot;]], &quot;)&quot;]], &quot;m&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <msup> <mi> z </mi> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8971; </mo> </mrow> </mrow> </mrow> </msup> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8805; </mo> <mi> m </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <ci> LegendreQ </ci> <ci> &#120083; </ci> </apply> <ci> n </ci> </apply> <ci> m </ci> </apply> <ci> z </ci> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> m </ci> <apply> <floor /> <apply> <times /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <ci> m </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <apply> <times /> <ci> m </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <ci> Binomial </ci> <ci> n </ci> <apply> <floor /> <apply> <times /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Binomial </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <floor /> <apply> <times /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <ci> n </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <floor /> <apply> <times /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <ci> m </ci> </apply> <apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <floor /> <apply> <times /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <ci> O </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <apply> <in /> <ci> m </ci> <ci> &#8469; </ci> </apply> <apply> <in /> <ci> n </ci> <ci> &#8469; </ci> </apply> <apply> <geq /> <ci> n </ci> <ci> m </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreP", "[", RowBox[List["n_", ",", "m_", ",", "3", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["m", "+", RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "m"]], "2"], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List["m", "/", "2"]]], " ", SuperscriptBox["2", RowBox[List["-", "n"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "m"]], "2"], "]"]]]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "n"]], "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "m"]], "2"], "]"]]]]]], ",", "n"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "m", "+", "n", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "m"]], "2"], "]"]]]]]], ",", "m"]], "]"]], " ", SuperscriptBox["z", RowBox[List["n", "-", "m", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "m"]], "2"], "]"]]]]]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", "z", "]"]]]], ")"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List["m", "/", "2"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", "0"]], ")"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List["n", "\[GreaterEqual]", "m"]]]]]]]]]]










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





2001-10-29





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