Zeros of weighted sum of two Bessel functions

Question

Just a simple and very tentative query to alleviate my seemingly futile internet digging: is there anything known on the structure of the entire function given by \begin{equation} f_c(z):=zJ_0'(z)+cJ_0(z)=\pm zJ_{\mp1}(z)+cJ_0(z)? \end{equation} In particular, any information on its zeros would be appreciated, though I am aware nothing may be known so far. Thanks!

Answer 1

$f_c(z) = 0$ iff $$c = \dfrac{z J_1(z)}{J_0(z)} = \dfrac{z^2}{2 - \dfrac{z^2}{4 - \dfrac{z^2}{6 - \dfrac{z^2}{8 - \ldots}}}}$$ (which might not help you much in computing $z$, but looks rather interesting).

Answer 2

A rather hand-wavy answer, but an answer nonetheless.

The zeros of any Bessel function accumulate at $+\infty$. This means that, for sufficiently large $n$, if $z_n$ denotes the $n$-th zero, then \begin{equation} J_1(z_n)=O(1/n). \end{equation} Thus we have, at least, \begin{equation} z_n=j_{1,n}+o(1). \end{equation} I will update this when I have acquired more precise asymptotics.