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Finite series within finite series

  1. Feb 4, 2010 #1
    1. The problem statement, all variables and given/known data
    I need to find a closed form for what at first light would be a straightforward finite series. Calculating it explicitly to a particular degree is not difficult, but I just can't find the closed form for the general case.

    2. Relevant equations
    For [tex]N>m[/tex], the series is:
    [tex]\sum_{k_m=1}^{N}(b^{m-1}x)^{k_m}\sum_{k_{m-1}=1}^{k_m}(b^{m-2}x)^{k_{m-2}}\cdots \sum_{k_2=1}^{k_3}(b x)^{k_2}\sum_{k_1=1}^{k_2}x^{k_2}[/tex]



    3. The attempt at a solution
    I know the answer will look something like this:
    [tex]\sum_{i=0}^{m}a_i(b,x) x^{i N}[/tex]
    where the [tex]a_i(b,x)[/tex] are linear combinations of b's and [tex]\frac{1}{1-(b^p x^r)^{-1}}[/tex], but I just can't find the general expression. Even a recurrence relation would be sufficient for my needs.

    Any help will be greatly appreciated.
     
  2. jcsd
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