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?



General solution to Laplacian eigenvalue equation in hyperbolic space
This seems to be addressed [in:](https://arxiv.org/abs/1201.4406)


*Cohl, H.S.; Kalnins, E.G.*, [**Fourier and Gegenbauer expansions for a fundamental solution of the Laplacian in the hyperboloid model of hyperbolic geometry**](http://dx.doi.org/10.1088/1751-8113/45/14/145206), J. Phys. A, Math. Theor. 45, No. 14, Article ID 145206, 32 p. (2012). [ZBL1238.31006](https://zbmath.org/?q=an:1238.31006).


