In the paper here, the authors present the equation
\begin{align} \nabla^2(\nabla^2 -\xi^{-2})\omega = 0 \end{align}
where $\xi$ is a constant and $\omega(r,\theta)$ is the variable of interest. This is equation (79) in the paper. In equation (82) they report
\begin{align} \nabla^2\omega(r,\theta) = \sum_{n=1}^\infty C_n K_n \left(\dfrac{r}{\xi}\right) \sin(n\theta) \end{align}
where $C_n$ is a constant and $K_n$ is the modified bessel function. They report the general form of $\omega$ as
\begin{align} \omega(r,\theta) = \sum_{n=1}^\infty \left[ a_{\omega,n}r^n+b_{\omega,n}r^{-n} + C_n\xi^rK_n \left(\dfrac{r}{\xi} \right) \right] \sin(n\theta) \end{align}
I am trying to reproduce their results and am unsure how they were able to determine this solution. If anyone could direct me towards any resources that discuss this type of equation that would be extremely helpful as well.