A note on zeros of derivatives of ultraspherical Bessel functions

Abstract

For any fixed $\nu \ge 0,\delta \in R$ and $x>0$, we investigate the positive zeros of the derivatives $j_{\nu ,\delta }^{\prime }\left(x\right)$ and $y_{\nu ,\delta }^{\prime }\left(x\right)$, where$j_{\nu ,\delta }\left(x\right)=x^{-\delta }{J}_{\nu }\left(x\right)\text{and}{y}_{\nu ,\delta }\left(x\right)=x^{-\delta }{Y}_{\nu }\left(x\right).$ We derive asymptotic expansions for their k-th positive zeros as $k\to \infty$.