A note on zeros of derivatives of ultraspherical Bessel functions

Abstract

For any fixed $\nu \ge 0,\delta \in \mathbb{R}$ and $x>0$, we investigate the positive zeros of the derivatives $j_{\nu ,\delta }^{\prime }(x)$ and $y_{\nu ,\delta }^{\prime }(x)$, 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 $k\to \infty$.