Bessel polynomial in terms of Bessel functions

Theorem: The following formula holds: $$y_n\left( \dfrac{1}{ir} \right) = \left(\dfrac{\pi r}{2} \right)^{\frac{1}{2}} e^{ir} \left[ \dfrac{J_{n +\frac{1}{2}}(r)}{i^{n+1}}+i^nJ_{-n-\frac{1}{2}}(r) \right],$$ where $y_n$ denotes a Bessel polynomial and $J_{\nu}$ denotes a Bessel function.

Proof: █