“Best possible” upper and lower bounds for the zeros of the Bessel function $J_\nu (x)$

Abstract:

Let $j_{\nu ,k}$ denote the $k$-th positive zero of the Bessel function $J_\nu (x)$. In this paper, we prove that for $\nu >0$ and $k=1$, 2, 3, $\ldots$, \[ \nu - \frac {a_k}{2^{1/3}} \nu ^{1/3} < j_{\nu ,k} < \nu - \frac {a_k}{2^{1/3}} \nu ^{1/3} + \frac {3}{20} a_k^2 \frac {2^{1/3}}{\nu ^{1/3}} . \] These bounds coincide with the first few terms of the well-known asymptotic expansion \[ j_{\nu ,k} \sim \nu - \frac {a_k}{2^{1/3}} \nu ^{1/3} + \frac {3}{20} a_k^2 \frac {2^{1/3}}{\nu ^{1/3}} + \cdots \] as $\nu \to \infty$, $k$ being fixed, where $a_k$ is the $k$-th negative zero of the Airy function $\operatorname {Ai}(x)$, and so are “best possible”.