How do you find the partial fraction decomposition of $$\frac{x^{2m-1}}{\prod_{k=1}^{n} (x^{2}+k^{2})} \, ,$$ where $m$ is some positive integer?
I don't know how to approach this other than to find the decomposition for different values of $m$ and $n$ and try to notice a pattern.
Is there a more systematic way I can approach this?
One standard trick is as follows. Write: $$ \frac{x^{2 m}}{\prod_{1 \le k \le n}(x^2 + k^2)} = \sum_{1 \le k \le n} \frac{a_k}{x - i k} + \frac{b_k}{x + i k} $$ Multiply through by $x - i k$, and make $x \to \dfrac{1}{i k}$ (here $i = \sqrt{-1}$). On the right hand side only the term for $a_k$ survives. With a bit of luck you can get the value of the left hand side. Similarly for $x + i k$ and $b_k$.