Fixed-argument growth of spherical Hankel functions ($h_n^{(1)}(t)\sim -i\sqrt{2}{t}^{-1}(2n/(et){)}^n$)

For fixed $t>0$ and large integer order, $y_n(t)\sim -(2n-1)!!/t^{n+1}$ while $j_n(t)$ is negligible. Stirling's formula gives $(2n-1)!!=(2n)!/(2^{n}n!)\sim \sqrt{2}(2n/e{)}^n$, proving the displayed asymptotic. In particular its magnitude is bounded above and below by constant multiples of $(2n/(et){)}^n$. This two-sided comparison is stronger than a one-sided big-O estimate and is what converts a finite spherical trace norm into a weighted condition on far-field coefficients.