I have to obtain an asymptotic solution for small real positive $x$ for the ratio of Spherical Hankel functions ($n=0,1,2....)$


${h^{(2)}\_n(x)}/{h^{(1)}\_n(x)}$


I found that series should be


$-1 + i C\_n x^{2n+1}+O(x^{2n+2})$


but how can I get the accurate coefficient at least for the first non-zero order?



The ratio of Hankel functions
$$\frac{h^{(2)}\_n(x)}{h^{(1)}\_n(x)}=-1+\frac{i \pi x^{2n+1}}{2^{2n}\Gamma \left(n+\frac{1}{2}\right) \Gamma \left(n+\frac{3}{2}\right)}+{\cal O}(x^{2n+2}).$$
For $n\geq 1$ the next term is actually of order $x^{2n+3}$.


