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variants of this functions
LegendreP






Mathematica Notation

Traditional Notation









Polynomials > LegendreP[n,mu,2,z] > Identities > Functional identities > Expansion with respect to parameters





http://functions.wolfram.com/05.07.17.0009.01









  


  










Input Form





LegendreP[Subscript[n, 1], Subscript[\[Mu], 1], z] LegendreP[Subscript[n, 2], Subscript[\[Mu], 2], z] == (-1)^Subscript[\[Mu], 1] Sqrt[((Subscript[n, 1] + Subscript[\[Mu], 1])! (Subscript[n, 2] + Subscript[\[Mu], 2])!)/ ((Subscript[n, 1] - Subscript[\[Mu], 1])! (Subscript[n, 2] - Subscript[\[Mu], 2])!)] Sum[If[k >= Abs[Subscript[\[Mu], 1] - Subscript[\[Mu], 2]] && EvenQ[k + Subscript[n, 1] + Subscript[n, 2]], (-1)^(-Subscript[\[Mu], 1] + Subscript[\[Mu], 2]) (2 k + 1) ThreeJSymbol[{Subscript[n, 1], 0}, {Subscript[n, 2], 0}, {k, 0}] ThreeJSymbol[{Subscript[n, 1], -Subscript[\[Mu], 1]}, {Subscript[n, 2], Subscript[\[Mu], 2]}, {k, Subscript[\[Mu], 1] - Subscript[\[Mu], 2]}] Sqrt[(k - Subscript[\[Mu], 2] + Subscript[\[Mu], 1])!/ (k + Subscript[\[Mu], 2] - Subscript[\[Mu], 1])!] LegendreP[k, -Subscript[\[Mu], 1] + Subscript[\[Mu], 2], z], 0], {k, Abs[Subscript[n, 1] - Subscript[n, 2]], Subscript[n, 1] + Subscript[n, 2]}] /; Element[Subscript[\[Mu], 1], Integers] && Element[Subscript[\[Mu], 2], Integers]










Standard Form





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MathML Form







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</ci> <cn type='integer'> 1 </cn> </apply> <ci> z </ci> </apply> <apply> <ci> LegendreP </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> &#956; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> &#956; </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <factorial /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> &#956; </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <factorial /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <apply> <abs /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </lowlimit> <uplimit> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> </uplimit> <apply> <ci> If </ci> <apply> <and /> <apply> <geq /> <ci> k </ci> <apply> <abs /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> &#956; 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</ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> LegendreP </ci> <ci> k </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> &#956; </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <ci> z </ci> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <apply> <ci> Subscript </ci> <ci> &#956; </ci> <cn type='integer'> 1 </cn> </apply> <ci> &#8484; </ci> </apply> <apply> <in /> <apply> <ci> Subscript </ci> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> <ci> &#8484; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2002-12-18





© 1998- Wolfram Research, Inc.