I'd like to approximate D[BesselK[n, x], n]/BesselK[n, x] and BesselK[n+1, x]/BesselK[n, x], knowning that n and x are both positive real numbers (e.g. ranging from 1.0 to 10^7). So if there is any analytic approximation for the BesselK function, the numerical overflow problem may be solved.
I find that the approximations in DLMF https://dlmf.nist.gov/10.30#E2 and https://dlmf.nist.gov/10.41#E2 are quite inaccurate (when there's no numerical overflow), for they are made for n->0 and n->Inf. I also read some journals on the lower and upper bound of BesselK, but it's an interval rather than an exact approximation, and I find it impossible to use iterative solutions in EM algorithm (note that n and x can be non-integers). Also, the concept of close to zero or Inf are not given (when n or x are larger or smaller than a threhold).
Could anyone please tell me how to approximate D[BesselK[n, x], n]/BesselK[n, x] and BesselK[n+1, x]/BesselK[n, x]? Or, is it possible to simplify the division of BesselK functions or derivative with the order n?