In my research the following equation appeared:

$$\frac{1}{4\pi}\int_{0}^{1}\frac{t^{s-1}(1-t)^{s-1}}{(\rho-t)^s}dt=\int_0^{\infty} f(a) Q^{i\sqrt{a}}_{s-1}(2\rho-1) da,$$

where $\rho,s>1$, $Q^{\mu}_{\nu}$ is the associeted Legendre function of second kind and $f(a)$ is to be found.

All my attempts to solve this equation failed. Maybe it is much to difficult to solve. However, I'm wondering if anyone knows similar equations appearing somewhere else in mathematics and if there are fruitful results regarding integrals similar to the right hand side with various $f(a)$? It looks somehow similar to an integral transform, e.g. like the Mehler Fock transform.

Best wishes