Fun and games with Mathematica
Here is some code to work out the asymptotic expansion of Bessel functions in the transition region. That is where the argument and order are both large, and the usual asymptotic series are inaccurate. The technique is from FWJ Olver in 1952. Olver worked out only the 5th order expansion, and no more terms seem to be in published papers. The NIST DLMF still only gives those terms from 1952. The code given here can compute it to arbitrary orders, and contains the result of computation of 28 terms.
Here is some code to work out the asymptotic expansion of the zeros of Bessel functions. The technique is from FWJ Olver in 1951. Olver worked out only 5th order for zeros of Bessel functions, and 4th order for zeros of derivatives of Bessel functions. The NIST DLMF still only gives those terms from 1951. The code given here can compute it to arbitrary orders, and contains the result of computation of another 19 terms.