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variants of this functions
LegendreP






Mathematica Notation

Traditional Notation









Polynomials > LegendreP[n,mu,2,z] > Series representations > Generalized power series > Expansions at z==0





http://functions.wolfram.com/05.07.06.0001.01









  


  










Input Form





LegendreP[n, \[Mu], 2, z] == Sum[(Pochhammer[-n, k] Pochhammer[n + 1, k] Pochhammer[\[Mu]/2 - k, m] Pochhammer[-\[Mu]/2, j] (-1)^j) (z^(j + m)/(Gamma[1 - \[Mu] + k] k! m! j! 2^k)), {k, 0, Infinity}, {m, 0, Infinity}, {j, 0, Infinity}]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["LegendreP", "[", RowBox[List["n", ",", "\[Mu]", ",", "2", ",", "z"]], "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "n"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["n", "+", "1"]], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["\[Mu]", "/", "2"]], "-", "k"]], ",", "m"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", "\[Mu]"]], "/", "2"]], ",", "j"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"]]], ")"]], RowBox[List[SuperscriptBox["z", RowBox[List["j", "+", "m"]]], "/", RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "k"]], "]"]], " ", RowBox[List["k", "!"]], RowBox[List["m", "!"]], RowBox[List["j", "!"]], SuperscriptBox["2", "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> <msubsup> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mi> n </mi> <mi> &#956; </mi> </msubsup> <mo> ( </mo> <semantics> <mi> z </mi> <annotation encoding='Mathematica'> TagBox[&quot;z&quot;, HoldComplete[LegendreP, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> m </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;-&quot;, &quot;n&quot;]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;n&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mfrac> <mi> &#956; </mi> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[FractionBox[&quot;\[Mu]&quot;, &quot;2&quot;], &quot;-&quot;, &quot;k&quot;]], &quot;)&quot;]], &quot;m&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mi> &#956; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;-&quot;, FractionBox[&quot;\[Mu]&quot;, &quot;2&quot;]]], &quot;)&quot;]], &quot;j&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mi> j </mi> <mo> + </mo> <mi> m </mi> </mrow> </msup> </mrow> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> &#956; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> m </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> j </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mi> k </mi> </msup> </mrow> </mfrac> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> LegendreP </ci> <ci> n </ci> <ci> &#956; </ci> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <sum /> <bvar> <ci> m </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <ci> k </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <ci> k </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <apply> <times /> <ci> &#956; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <ci> m </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> &#956; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> j </ci> </apply> <apply> <power /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> j </ci> <ci> m </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <factorial /> <ci> k </ci> </apply> <apply> <factorial /> <ci> m </ci> </apply> <apply> <factorial /> <ci> j </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </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_", ",", "\[Mu]_", ",", "2", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "n"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["n", "+", "1"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["\[Mu]", "2"], "-", "k"]], ",", "m"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["\[Mu]", "2"]]], ",", "j"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"]]], ")"]], " ", SuperscriptBox["z", RowBox[List["j", "+", "m"]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "k"]], "]"]], " ", RowBox[List["k", "!"]], " ", RowBox[List["m", "!"]], " ", RowBox[List["j", "!"]], " ", SuperscriptBox["2", "k"]]]]]]]]]]]]]]










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





2001-10-29





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