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variants of this functions
LegendreP






Mathematica Notation

Traditional Notation









Hypergeometric Functions > LegendreP[nu,mu,2,z] > Series representations > Generalized power series > Expansions at z==infinity





http://functions.wolfram.com/07.08.06.0040.01









  


  










Input Form





LegendreP[\[Nu], \[Mu], 2, z] == (((-1)^(2 \[Nu] + 1) 2^(\[Nu] + 1))/((-\[Mu] - \[Nu] - 1)! Gamma[-\[Nu]])) (z - 1)^(-\[Nu] - 1) ((1 + z)^(\[Mu]/2)/(1 - z)^(\[Mu]/2)) Sum[((Pochhammer[\[Nu] + 1, k] Pochhammer[\[Mu] + \[Nu] + 1, k])/ (k! (2 \[Nu] + k + 1)!)) (Log[(z - 1)/2] + PolyGamma[k + 1] - PolyGamma[-k - \[Mu] - \[Nu]] - PolyGamma[\[Nu] + k + 1] + PolyGamma[2 \[Nu] + k + 2]) (2/(1 - z))^k, {k, 0, -\[Mu] - \[Nu] - 1}] + ((2^\[Nu] Gamma[1/2 + \[Nu]])/ (Sqrt[Pi] Gamma[1 - \[Mu] + \[Nu]])) (z - 1)^\[Nu] ((1 + z)^(\[Mu]/2)/(1 - z)^(\[Mu]/2)) Sum[((Pochhammer[\[Mu] - \[Nu], k] Pochhammer[-\[Nu], k])/ (k! Pochhammer[-2 \[Nu], k])) (2/(1 - z))^k, {k, 0, 2 \[Nu]}] + (((-1)^(2 \[Nu]) 2^(1 - \[Mu]) Sin[Pi \[Nu]] Gamma[1 - \[Mu]])/ (Pi Gamma[1 - \[Mu] - \[Nu]] Gamma[2 - \[Mu] + \[Nu]])) (z - 1)^(\[Mu] - 1) ((1 + z)^(\[Mu]/2)/(1 - z)^(\[Mu]/2)) HypergeometricPFQ[{1, 1, 1 - \[Mu]}, {1 - \[Mu] - \[Nu], 2 - \[Mu] + \[Nu]}, 2/(1 - z)] /; Element[2 \[Nu] + 1, Integers] && 2 \[Nu] + 1 >= 0 && Element[\[Nu] - \[Mu], Integers] && \[Mu] + \[Nu] <= 0










Standard Form





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MathML Form







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</mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </munderover> <mrow> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;\[Mu]&quot;, &quot;-&quot;, &quot;\[Nu]&quot;]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;-&quot;, &quot;\[Nu]&quot;]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> </mrow> <mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; 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Rule Form





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





2001-10-29





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