I was able to prove that if $n \in \mathbb{N}$ then $K_n(\rho)$ has the following series representation
$$K_n(\rho) = 1/2\sum_{m=0}^{n-1}\frac{(-1)^m (n-m-1)!}{m!}(\rho/2)^{2m-n}+(-1)^{n-1}/2 \sum_{m=0}^{\infty}\frac{(\rho/2)^{2m+n}}{m!(m+n)!}[2\log(\rho/2)-\psi(m+1)-\psi(m+n+1) ]$$
In the Book by N.N Lebedev and Richard R. Silverman called special functions and their applications, page 111, they say that this series representation for $K_n$ imply that when $\rho \to 0$
$$K_0(\rho)\approx \log(2/\rho)$$ $$K_n(\rho) \approx\frac{1}{2}(n-1)!(\rho/2)^{-n}\,\,\,\text{for}\,\,\, n=1,2,\dots, $$
But the book does not give the proof, and I'm trying and trying to prove this without success, I need help in finding which result the authors use to find this behaviors near zero so easily that does not even deserve a proof, I think that the behaviors above does not follow by simply calculation. Any reference on how to justify the above will be very helpful, thank you so much for the patience to read.