General solution to Laplacian eigenvalue equation in hyperbolic space

Question

In hyperbolic space $H^n$ with metric $g=dr^2+\sinh^2 r\:g_{S^{n-1}}$, consider the Laplacian eigenvalue equation

$-\Delta u=c^2 u$.

I am concerned with the radial solution to the above equation, $u=u(r)$. So the equation becomes

$$v_{rr}+(n-1)\coth r\cdot u_r+c^2u=0.$$

What is the general solution to the equation above?

If we set $u(r)=\varphi(s)=\varphi(\frac{1-\cosh r}{2})$, we can get

$$ s(1-s)\varphi_{ss}+(\frac{n}{2}-ns)\varphi_s-c^2\varphi=0.$$

It is known that the solution is given by hypergeomertic functions. What is the general solution which is regular at the origin? Are there two independent solutions? Is the case $n=2$ simpler?

Answer 1

This seems to be addressed in:

Cohl, H.S.; Kalnins, E.G., Fourier and Gegenbauer expansions for a fundamental solution of the Laplacian in the hyperboloid model of hyperbolic geometry, J. Phys. A, Math. Theor. 45, No. 14, Article ID 145206, 32 p. (2012). ZBL1238.31006.