= Fixed-argument growth of spherical Hankel functions
{title2=$h_n^{(1)}(t)\sim-i\sqrt2\,t^{-1}(2n/(et))^n$}
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 $(2n-1)!!=(2n)!/(2^nn!)\sim\sqrt2(2n/e)^n$, 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.
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