Zeros of Bessel function derivatives

TL;DR

This paper proves that for certain parameters, all zeros of the derivatives of Bessel functions are real and simple, and establishes interlacing properties of these zeros, extending classical results and proposing open problems.

Contribution

It introduces new proofs and results on the zeros of derivatives of Bessel functions, including interlacing and reality, using advanced complex analysis techniques.

Findings

Zeros of derivatives are real and simple for specified parameters.

Positive zeros of consecutive derivatives interlace when parameters meet certain conditions.

Extends classical results and proposes open problems on Bessel function zeros.

Abstract

We prove that for $\nu>n-1$ all zeros of the $n$th derivative of Bessel function of the first kind $J_{\nu}$ are real and simple. Moreover, we show that the positive zeros of the $n$th and $(n+1)$th derivative of Bessel function of the first kind $J_{\nu}$ are interlacing when $\nu\geq n,$ and $n$ is a natural number or zero. Our methods include the Weierstrassian representation of the $n$th derivative, properties of the Laguerre-P\'olya class of entire functions, and the Laguerre inequalities. Some similar results for the zeros of the first and second derivative of the Struve function of the first kind $\mathbf{H}_{\nu}$ are also proved. The main results obtained in this paper generalize and complement some classical results on the zeros of Bessel functions of the first kind. Some open problems related to Hurwitz theorem on the zeros of Bessel functions are also proposed, which may be of interest for further research.