Bounds and monotonicities for the zeros of derivatives of ultraspherical Bessel functions

The positive zeros $p_{\nu k}^{(\ell )} $ of $[x^{ - \nu + 1} J_{\nu + \ell - 1} (x)]'$, $\nu + \ell > 0$, where $J_\nu (x)$ denotes the standard Bessel function, arise in the study of the eigenvalues of Neumann Laplacians in N dimensions [M. S. Ashbaugh and R. D. Benguria, SIAM J. Math. Anal., 24 (1993), pp. 557–570]. The case $\ell = 1$ is particularly relevant. To pave the way for these applications, the authors present here inter alia (i) lower and upper bounds for $p_{\nu 1}^{(\ell )} $ and (ii) an explicit representation for ${{dp_{\nu k}^{(\ell )} } /{d\nu }}$. The latter implies that $p_{\nu k}^{(\ell )} $ is increasing in $\nu $ for fixed $k,\ell $, provided $\nu + \ell > 1$.