Closed form for sum of a simple power law series

Question

I'm struggling to find a closed form for $g_{ij} = \sum_{k=1}^{N-1} \frac{1}{k^2}10^{-2dk}i^kj^k$ where $i, j, d, k, N$ are constant natural numbers. This, of course, isn't a geometric series due to the power law term $\frac{1}{k^2}$ at the beginning (if it were, I'd just use the finite geometric sum formula), so I'm not sure how to proceed, but it seems reasonably simple, so I wonder if it has a closed form.

Answer 1

Let $x=10^{-d}ij$. So $f(x)=\sum\limits_{k=1}^{N-1}\frac{x^k}{k^2}$. and $f'(x)=\sum\limits_{k=1}^{N-1}\frac{x^{k-1}}{k}$. So (using $xf'(x))$ $f'(x)+xf''(x)=\sum\limits_{k=1}^{N-1}x^{k-1}=\frac{1-x^{N-1}}{1-x}$.

Solve diff. equation to get $f(x)$.

Answer 2

The solution of the differential equation $$f'(x)+xf''(x)=\frac{1-x^{n-1}}{1-x}$$ is not bad $$f(x)=\text{Li}_2(x)+c_1 \log (x)+c_2-x^{n+1} \Phi (x,2,n+1)-\frac{x^n}{n^2}$$ where $\Phi(.)$ is the Hurwitz-Lerch transcendent function which is available in many scientific libraries.

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