Bounds on differences of adjacent zeros of Bessel functions and iterative relations between consecutive zeros

Bounds for the distance $\vert c_{\nu ,s}-c_{\nu \pm 1 , s^{\prime}}\vert$ between adjacent zeros of cylinder functions are given; $s$ and $s^{\prime}$ are such that $\nexists c_{\nu,s^{\prime\prime}}\in ]c_{\nu ,s},c_{\nu\pm 1,s^{\prime}}[$; $c_{\nu ,k}$ stands for the $k$th positive zero of the cylinder (Bessel) function $\mathcal{C}_{\nu}(x)=\cos\alpha J_{\nu}(x) - \sin\alpha Y_{\nu}(x)$, $\alpha \in [0,\pi[$, $\nu \in {\mathbb R}$.

These bounds, together with the application of modified (global) Newton methods based on the monotonic functions $f_{\nu}(x)=x^{2\nu -1}\mathcal{C}_{\nu}(x)/\mathcal{C}_{\nu -1}(x)$ and $g_{\nu}(x)=-x^{-(2\nu +1)}\mathcal{C}_{\nu}(x)/\mathcal{C}_{\nu +1}(x)$, give rise to forward ( $c_{\nu ,k} \rightarrow c_{\nu ,k+1}$) and backward ( $c_{\nu ,k+1} \rightarrow c_{\nu ,k}$) iterative relations between consecutive zeros of cylinder functions.

The problem of finding all the positive real zeros of Bessel functions $\mathcal{C}_{\nu}(x)$ for any real $\alpha$ and $\nu$ inside an interval $[x_{1},x_{2}]$, $x_{1}>0$, is solved in a simple way.