Bessel function at large order I'm trying to expand a modified Bessel function such that
$$K_n \left(\sqrt{n} \left(a_0 + a_1 \frac{1}{n} + a_2 \frac{1}{n^2} + \ldots \right)\right) = A(n) \left( b_0 + b_1 \frac{1}{n} + b_2 \frac{1}{n^2} + \ldots \right) $$
in the large $n$ limit and $A(n)$ is some function of $n$ that contains diverging part. Is such expansion possible? How would one obtain the coefficients $b_0, b_1, \ldots$ in terms of $a_0, a_1, \ldots$?
I would appreciate any hint or reference. Thank you.

Unless I misunderstand, there's something missing here.  It looks to me like $K_n(\sqrt{n})$ blows up rather rapidly as $n \to \infty$.  For example, 
for $n=10$ I get approximately $1413.798936$, for $n = 20$ approximately $4.796177691 \times 10^9$.
