Asymptotic behavior of zeros of Bessel function derivatives

Abstract

We derive two distinct asymptotic expansions for the zeros $j_{\nu,k}^{(n)}$ of the $n$-th derivative of Bessel function $J_\nu^{(n)}(x)$. The first is a McMahon-type expansion for the case when $k \to \infty$ with fixed $\nu$, for which we also establish an explicit error bound. The second addresses the case when $\nu \to \infty$ with fixed $k$ and it involves the zeros of Airy functions and their derivatives. These results extend and refine the classical work of Wong, Lang, and Olver on the zeros of Bessel functions. In the course of obtaining our main results, we also generalize several auxiliary results, which in turn provide a broader framework for the study of zeros of special functions.