Looking at the roots of the Bessel functions (particularly the Bessel functions of the first kind, $J_\nu(x)$), it seems that a lot is known. Poking around WolframAlpha plots, I have observed a couple of patterns in the spacing of adjacent roots, and I wanted to know if they've been proven in general or not. If $j_{\nu,n}$, $n \ge 1$, is the $n^{\mathrm{th}}$ positive zero of $J_\nu(x)$, then $\forall\ |\nu| \ge \frac{1}{2}$: \begin{align} j_{\nu,n+1}-j_{\nu,n} &\ge \pi \\ j_{\nu,n+1}-j_{\nu,n} &\ge j_{\nu,n+2}-j_{\nu,n+1}, \end{align} with the inequalities reversing for $|\nu| < \frac{1}{2}$, but not by much.
The reference linked in an answer to a previous question about Bessel function roots has on the bottom of page 492 (502 in pdf numbering) that for functions of the form $$\mathcal{C}_\nu(x) \equiv J_\nu(x) \cos\alpha - Y_\nu(x) \sin\alpha$$ with $\nu > \frac{1}{2}$ the roots that exceed $\frac{(2\nu+1)(2\nu+3)}{\pi}$ are bounded by the intervals $$\left(m\pi - \alpha + \frac{\nu\pi}{2} + \frac{\pi}{2},\ m\pi - \alpha + \frac{\nu\pi}{2} + \frac{3\pi}{4}\right)$$ for some integers $m$ (note that I cannot find any guarantee that the $m$ in this formula will correspond to the $n$ indexing the zeros, overall).
The limiting form of $J_\nu(x)$ certainly implies that the spacing has to trend to $\pi$ as $n\rightarrow \infty$, I know of no proof about whether that trend is from above or below.
Sorry, I'm rambling because it's late. Point being - are these properties provable?
