Is there a version of Bessel functions for hyperbolic sectors?

Question

I would like to compute the Neumann eigenfunctions of a sector in the hyperbolic plane with angle $\beta$. Working in the Poincare disk, we can employ separation of variables using polar coordinates. The equation for the radial component becomes $$r^2f''+rf'+\Big(\frac{4\lambda r^2}{(1-r^2)^2}-\frac{n^2\pi^2}{\beta^2}\Big)f=0,$$ where $\lambda$ is the eigenvalue we're solving for. Using the Frobenius method (see e.g. Teschl's Ordinary Differential Equations and Dynamical Systems), we can construct a solution that is a power series of an analytic function multiplied by a (potentially non-integer) power of $r$. This series shares the property of the Bessel function that all the terms in the power series consist only of even powers of $r$. Does anyone know if this function has been discussed anywhere in the literature and if it has a name? I have not been able to find it in the standard references on special functions, though I may have seen it and just not recognized it as such.