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This summation sums to zero. Why?

  1. Feb 5, 2012 #1
    Hi,

    I am reading a paper, and at some point the authors claim that:

    [tex]\sum_{m=1}^{L+1}\frac{\prod_{\substack{l=1\\l\neq m}}^{L+1}\frac{\lambda(m)}{\lambda(m)-\lambda(l)}}{\lambda^r(m)}=0[/tex]

    the question is HOW?

    Any tiny hint will be highly appreciated.

    Thanks
     
  2. jcsd
  3. Feb 8, 2012 #2
    In general there is no equality.
    It must depend on the definitions of λ, r and L.
    Can you provide more details?
     
  4. Feb 8, 2012 #3
    Lambdas are positive numbers, r is between 1 and L. That is all
     
  5. Feb 8, 2012 #4
    I don't think that is correct.

    Define [itex]\lambda(m)=m[/itex], and pick [itex]L=r=2[/itex]. Then
    [tex]\frac{\frac{\lambda(1)}{\left(\lambda(1)-\lambda(2)\right)\left(\lambda(1)-\lambda(3)\right)}}{\lambda(1)^2}+\frac{\frac{\lambda(2)}{\left(\lambda(2)-\lambda(1)\right)\left(\lambda(2)-\lambda(3)\right)}}{\lambda(2)^2}+\frac{\frac{\lambda(3)}{\left(\lambda(3)-\lambda(1)\right)\left(\lambda(3)-\lambda(2)\right)}}{\lambda(3)^2}=\frac{1}{2}-\frac{1}{2}+\frac{1}{6}=\frac{1}{6}[/tex]
     
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