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## Homework Statement

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.

## Homework 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]

## 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.