My attempts to search via Google seem to be failing, so I thought of asking here.
All the derivatives of the function
$r_n(z):=\frac{J_n(z)}{J_{n-1}(z)}$
where $J_n(z)$ is the Bessel function of the first kind 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 (linear?) 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:
- Are there any papers where generating functions/differential equations of ratios of Bessel functions have been studied? ; or
- How can I derive a differential equation for $r_n(z)$ with the knowledge that all higher derivatives are expressible in terms of $r_n(z)$?
(On the other hand, the difference equation for $r_n(z)$ (and thus its continued fraction representation) is easily derived, so no problem for me there.)
I will be interested in any input. Thanks!
EDIT:
Per Pietro's request, I now tip my hand and reveal my reason for interest: I saw this paper many years ago on a neat method for computing the first few roots of the Bessel function of the first kind. Some time later, I came across J.F. Traub's "Iterative Methods for the Solution of Nonlinear Equations", where he shows the construction of iteration functions involving derivatives and can be constructed to have quadratic, cubic... convergence. (Newton's method is but the first member of this family). I also came across this short note by D.J. Hofsommer on how one might profitably exploit the methods derived by Traub if the function of interest satisfies a simple differential equation (Essentially, one just constructs the Newton correction $u=\frac{f(z)}{f^{\prime}(z)}$, and the high-order iteration functions are merely a series in powers of $u$). That got me wondering on how one might recursively generate iteration functions with increasing order of convergence for the case of finding the roots of the Bessel function. (On another note, I was able to successfully use the ideas of Traub and Hofsommer for the generation of Gaussian quadrature rules, e.g. Legendre, Lobatto, Radau, and was hoping things might be just as successful for Bessel function root-finding).