Monotonicity of zeros of derivatives of Bessel functions

Abstract

Recently Baricz et al., 2018 and Baricz and Singh 2018 gave two different proofs of the fact that the zeros of the $n$th derivative of the Bessel function of the first kind $J_{\nu }\left(x\right)$ are all real when $\nu >n-1$. We provide a third alternative proof. The authors of Baricz et al., 2018 conjectured that, for every $n\in N$, the positive zeros of $J_{\nu }^{\left(n\right)}\left(x\right)$ are increasing functions of the parameter $\nu$, for $\nu \in (n-1,\infty )$. We provide two apparently distinct proofs of the conjecture.