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variants of this functions
LegendreQ






Mathematica Notation

Traditional Notation









Hypergeometric Functions > LegendreQ[nu,mu,2,z] > Differentiation > Symbolic differentiation > With respect to nu





http://functions.wolfram.com/07.11.20.0011.02









  


  










Input Form





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










Standard Form





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MathML Form







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</mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </munderover> <mrow> <msubsup> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox[&quot;S&quot;, StirlingS1] </annotation> </semantics> <mi> k </mi> <mrow> <mo> ( </mo> <mi> i </mi> <mo> ) </mo> </mrow> </msubsup> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> i </mi> <mo> - </mo> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;i&quot;, &quot;-&quot;, &quot;j&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;)&quot;]], &quot;j&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <msup> <mi> &#957; </mi> <mrow> <mi> i </mi> <mo> - </mo> <mi> j </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> r </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> r </mi> </msup> <mo> &#8290; 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</ci> <pi /> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> z </ci> </apply> <apply> <times /> <ci> &#956; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <ci> &#956; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <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> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> k </ci> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <times /> <apply> <ci> Binomial </ci> <ci> m </ci> <ci> j </ci> </apply> <apply> <sum /> <bvar> <ci> i </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <times /> <apply> <ci> StirlingS1 </ci> <ci> k </ci> <ci> i </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> i </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <cn type='integer'> 1 </cn> </apply> <ci> j </ci> </apply> <apply> <power /> <ci> &#957; 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Rule Form





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





2001-10-29





© 1998- Wolfram Research, Inc.