The following sums pop up in diffraction theory and are related to Lommel's function of two variables. Let $u,v\in\mathbb{R}$. I claim that $$\sum_{n=0}^\infty i^n \left ( \frac{u}{v} \right )^n J_{n+1}(v)=\text{sign}(u)e^{iu/4}\sqrt{\frac{2\pi}{u}}\sum_{n=0}^\infty i^n (2n+1)J_{n+1/2}\left ( \frac{u}{4} \right )J_{2n+1}(v),$$ where $J_a(b)$ is the Bessel J.
Can anyone show this?
The left hand side is related to Lommel's original definition and the right hand side was derived by Zernike and Nijboer in 1947 (published 1949). However, Zernike and Nijboer are light on detail.
The expansion on the RHS may somehow be related to the Bauer/Rayleigh expansion: $$e^{ikr\cos\theta}=\sum_{n=0}^\infty i^n(2n+1)j_n(kr)P_n(\cos\theta),$$ where $j_n$ is the spherical Bessel j and $P_n$ is the $n$th Legendre polynomial.
Also, I added the $\text{sign}(u)$ myself in order to make things fit numerically. The expression published by Boersma (where I got the expressions) is missing the signum and doesn't seem to work out for negative u.