Ramanujan’s Incorrect Partial Fraction Expansion and Ramanujan’s Celebrated Formula for $\zeta(2n + 1)$

Question

I was reading the article An Unpublished Manuscript of Ramanujan on Infinite Series Identities of G.E. Andrews and B.C. Berndt, Ramanujan's Lost Notebook and I wondered if you could get a partial fraction decomposition of the function $$\frac{1}{2} \pi \cot \left(\sqrt{\alpha z}\right) \coth \left(\sqrt{\beta z}\right)=\sum _{m=1}^{\infty } \left(\frac{\alpha m \coth (\alpha m)}{\alpha m^2+z}+\frac{\beta m \coth (\beta m)}{z-\beta m^2}\right)+\frac{1}{2} \log \left(\frac{\beta }{\alpha }\right)+\frac{1}{2 z}$$ for any value of $$\beta ,\alpha $$ not just for values $$\alpha \beta =\pi ^2$$ or maybe it can't, they could explain to me why those restrictions