Finite series within finite series

[al2521300]

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 $$N>m$$, the series is:
$$\sum_{k_m=1}^{N}(b^{m-1}x)^{k_m}\sum_{k_{m-1}=1}^{k_m}(b^{m-2}x)^{k_{m-2}}\cdots \sum_{k_2=1}^{k_3}(b x)^{k_2}\sum_{k_1=1}^{k_2}x^{k_2}$$

The Attempt at a Solution

I know the answer will look something like this:
$$\sum_{i=0}^{m}a_i(b,x) x^{i N}$$
where the $$a_i(b,x)$$ are linear combinations of b's and $$\frac{1}{1-(b^p x^r)^{-1}}$$, 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.

 

[mighty2000]

Thank you for reaching out for assistance with your series problem. I understand the frustration of not being able to find a closed form solution for a seemingly straightforward series. However, with some perseverance and the right tools, I believe we can find a solution for your specific case.

After examining your series and attempting some calculations, I agree that the general expression for your series will likely involve linear combinations of b's and \frac{1}{1-(b^p x^r)^{-1}}. One approach you can try is to use generating functions, which can often help in finding closed form expressions for series. Alternatively, you can also try using recurrence relations, as you suggested, to simplify the problem and potentially find a solution.

If you are still struggling to find a solution, I would recommend consulting with a colleague or a mentor who may have more experience with similar problems. Sometimes, a fresh perspective can help in finding a solution. Additionally, there are also online communities and forums where you can seek help from other scientists and mathematicians.

I wish you the best of luck in finding a solution for your series. Keep persevering and don't hesitate to reach out for assistance when needed. As scientists, we are always learning and solving problems, and I have no doubt that you will eventually find a solution for your series.