A note on zeros of derivatives of ultraspherical Bessel functions

Abstract

For any fixed $ν\ge 0, δ\in \mathbb R$ and $x>0$, we investigate the positive zeros of the derivatives $j'_{ν,δ}(x)$ and $y'_{ν,δ}(x)$, where \begin{equation*} j_{ν,δ}(x)=x^{-δ}J_ν(x)\quad\text{and} \quad y_{ν,δ}(x)=x^{-δ}Y_ν(x). \end{equation*} We derive asymptotic expansions for their $k$-th positive zeros as $k\rightarrow \infty$.