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

Abstract:

Bounds for the distance $|c_{\nu ,s}-c_{\nu \pm 1 , s’ }|$ between adjacent zeros of cylinder functions are given; $s$ and $s’$ are such that $\nexists c_{\nu ,s^{\prime \prime }}\in \ ]c_{\nu ,s},c_{\nu \pm 1,s’ }[$; $c_{\nu ,k}$ 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.