Spherical Hankel function ($h_n^{(1)}=j_n+i{y}_n,h_n^{(2)}=j_n-i{y}_n$)

These combinations of Spherical Bessel functions solve the radial Helmholtz equation. For time dependence $e^{-i\omega t}$, $h_n^{(1)}(kr)$ is outgoing and $h_n^{(2)}(kr)$ incoming. At fixed order and large argument, $h_n^{(1)}(t)\sim (-i{)}^{n+1}{e}^{it}/t$. Their recurrence is $h_{n+1}=(2n+1)h_n/t-h_{n-1}$.

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.