A note on zeros of derivatives of ultraspherical Bessel functions

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