“Best possible” upper and lower bounds for the zeros of the Bessel function 𝐽_{𝜈}(𝑥)

Abstract

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