Let $J_\nu(x):=\displaystyle\sum^\infty_{k=0}\frac{(-1)^k(x/2)^{\nu+2k}}{k!~\Gamma(\nu+k+1)}$ denote a Bessel function. When $\nu\geq0$, let $0<j_{\nu,1}<j_{\nu,2}<\cdots$ denote the positive zeroes of $J_\nu(x)$. From this MSE post, one gets an asymptotic form of $j_{\nu, k}$ for large $k$ and fixed $\nu$ (see also MacMahon's asymptotic formula here). My question is:
Keeping $k$ fixed, are there asymptotic formulae (or lower bounds) for $j_{\nu, k}$ for large positive $\nu$?
Any help is highly appreciated, thanks!
