My attempts to search via Google seem to be failing, so I thought of asking here.

All the derivatives of the function

$r_n(z):=\frac{J_n(z)}{J_{n-1}(z)}$

are expressible in terms of $r_n(z)$, for instance $\frac{\mathrm{d}}{\mathrm{d}z}r_n(z)=r_n(z)^2-\frac{2n-1}{z}r_n(z)+1$ . I've been trying to derive a differential equation (hopefully just second-order) that might be satisfied by $r_n(z)$, but my manipulative ability does not seem to be up to snuff.

Probably my problem can be resolved in two ways:

1. Are there any papers where generating functions/differential equations of ratios of Bessel functions have been studied? ; or
2. How can I derive a differential equation for $r_n(z)$ with the knowledge that all higher derivatives are expressible in terms of $r_n(z)$ ?

I will be interested in any input. Thanks!