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variants of this functions
LegendreQ






Mathematica Notation

Traditional Notation









Hypergeometric Functions > LegendreQ[nu,mu,3,z] > Differentiation > Low-order differentiation > With respect to nu





http://functions.wolfram.com/07.12.20.0005.01









  


  










Input Form





D[LegendreQ[\[Nu], \[Mu], 3, z], {\[Nu], 2}] == (Pi/2) Csc[\[Mu] Pi] E^(Pi I \[Mu]) (((z + 1)^(\[Mu]/2)/(z - 1)^(\[Mu]/2)) Sum[(1/(Gamma[1 - \[Mu] + k] k!)) ((1 - z)/2)^k Sum[\[Nu]^(-2 + i) StirlingS1[k, i] Sum[(-1)^r (1 + \[Nu])^(-2 + r) ((r - 1) r \[Nu]^2 + i^2 (1 + \[Nu])^2 + i (1 + \[Nu]) ((2 r - 1) \[Nu] - 1)) StirlingS1[k, r], {r, 1, k}], {i, 1, k}], {k, 0, Infinity}] - Pochhammer[1 - \[Mu] + \[Nu], 2 \[Mu]] (((PolyGamma[1 - \[Mu] + \[Nu]] - PolyGamma[1 + \[Mu] + \[Nu]])^2 - PolyGamma[1, 1 - \[Mu] + \[Nu]] + PolyGamma[1, 1 + \[Mu] + \[Nu]]) LegendreP[\[Nu], -\[Mu], 3, z] + ((z - 1)^(\[Mu]/2)/(z + 1)^(\[Mu]/2)) Sum[(1/(Gamma[1 + \[Mu] + k] k!)) ((1 - z)/2)^k Sum[StirlingS1[k, j] \[Nu]^j Sum[(-1)^r StirlingS1[k, r] (1 + \[Nu])^r ((1/(\[Nu]^2 (1 + \[Nu])^2)) ((-1 + r) r \[Nu]^2 + j^2 (1 + \[Nu])^2 + j (1 + \[Nu]) (-1 + (-1 + 2 r) \[Nu])) + 2 (j/\[Nu] + r/(1 + \[Nu])) (-PolyGamma[1 - \[Mu] + \[Nu]] + PolyGamma[1 + \[Mu] + \[Nu]])), {r, 1, k}], {j, 1, k}], {k, 0, Infinity}])) /; Abs[(1 - z)/2] < 1 && !Element[\[Mu], Integers]










Standard Form





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MathML Form







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</mi> <mo> - </mo> <mi> &#956; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;\[Nu]&quot;, &quot;-&quot;, &quot;\[Mu]&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;)&quot;]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;\[Mu]&quot;]]], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> &#956; </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> &#956; </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mi> &#915; 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Date Added to functions.wolfram.com (modification date)





2001-10-29





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