In this 1943 paper by Hudson on the theory of elastic waves in beams the author offers an interesting way to calculate a function defined as
$$ \theta_n (z)=\frac{z J'_n (z)}{J_n (z)} $$
with $z$ either real or imaginary, $J_n(z)$ - Bessel function, and $'$ denoting $z$ derivative.
To simplify the computation of this function he offers a continued fraction expansion for it based on the recurrence relation for Bessel functions.
$$ \theta_n (z)=n-\frac{z^2}{2(n+1)-\frac{z^2}{2(n+2)-...}} $$
So my question - is there a theory about expansion of functions into continued fractions and what kinds of functions can be expressed this way? And what about the interval (ring) of convergence for such fractions?
Edit: This answer offers a lot of information.