Zeros of Bessel function derivatives

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 $\nu \ge n,$ and $n$ is a natural number or zero. Our methods include the Weierstrassian representation of the $n$th derivative, properties of the Laguerre-Pólya 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.

10 pages

Cited by in corpus (5)

• Bounds for radii of starlikeness of some $q$-Bessel functions
• Zeros of some special entire functions
• Bounds for radii of convexity of some $q$-Bessel functions
• When does a hypergeometric function ${}_p{F}_q$ belong to the Laguerre--Pólya class $L{P}^{+}$?
• Applications of third order differential subordination and superordination involving generalized Struve function