Series produces incomplete expansion for these Bessel functions?

Questions
1

Hi, I have again a problem with using Series in connection with Bessel functions. I need an expansion

F[s_]:=(BesselK[1,Sqrt[s]]/(Sqrt[s]*BesselK[0,Sqrt[s]]))/(1+th*Sqrt[d]*
BesselK[1,Sqrt[s]]*BesselK[0,Sqrt[d*s]]/BesselK[0,Sqrt[s]]/BesselK[1,Sqrt[d*s]]);

ser0=Series[F[s],{s,∞,1}]

Unfortunately the above command produces a clearly incompletely performed expansion, since the successive expansion coefficients are functions of s. I have reasons to expect that this is not correct, and that the correct complete expansion contains terms in the form (1/Sqrt[s])^n with coefficients independent of s. In fact, if we perform

Series[SeriesCoefficient[ser0,1/2],{s,∞,1}]

we can see that the first term can still be expanded. Is there any way to obtain a complete expansion, ideally up to 40 terms? Why Series exhibits such a weird behaviour?

Lesław

2

If I am not mistaken, the issue involves presence of Sqrt[d*s] that messes up the coefficients. This can be addressed by placing an assumption on s that it is strictly real-valued.

ser0 = 
 Series[F[s], {s, \[Infinity], 1}, Assumptions -> s > 1000]

(* Out[1497]= SeriesData[s, 
DirectedInfinity[
 1], {(1 + d^Rational[1, 2] th)^(-1), 0, 
  Rational[1, 2] (1 + th) (1 + d^Rational[1, 2] th)^(-2)}, 2, 6, 4] *)

Obtaining high degree will not be easy. This appears to blow up in size, in part due to intrinsic complexity of the result and in part from Series using less intermediate simplification than is needed for this example. It’s a tricky balancing act since such simplifying can be slow.

3

This does not seem to help in the case of MATHEMATICA 10 or 11.

Lesław