I am looking for the cylindrical wave expansions of spherical waves. For example, for zeroth order we have the Sommerfeld identity:
$\frac{e^{jkr}}{r}=\frac{j}{2}\int_{-\infty}^{\infty}\frac{e^{jk_z|z|}}{k_z}H_0^{(1)}(k_\rho\rho)k_\rho dk_\rho$
$k^2=k_\rho^2+k_z^2$ ; and $r^2=\rho^2+z^2$
I want to expand it to higher orders. Something along the line of:
$h_n^{(1)}(kr) = \int_{-\infty}^{+\infty} fH_n^{(1)}(k_\rho \rho)k_\rho^{n+1}dk_\rho $
with an unknown $f$ that I want to know. I am from electrical engineering, and this problem is relevant for finding the response of stratified media. My struggle is, the Sommerfeld identity was derived not through the mathematical properties of the functions, but by finding the Green's function of the scalar Helmholtz equation with two different approaches, and then applying uniqueness theorem. Since the higher orders are not the Green's function, I am not sure how to approach this problem.
Any solution or idea or reference to relevant literature is appreciated. Thanks!