I need to solve the following equation for $\lambda\in\mathbb{C}$: $$\alpha+\frac{\lambda}{4\pi}+\frac{i}{4}\sum_{n=-\infty}^{+\infty}\frac{J_{n}(iR\lambda)}{H_{n}^{(1)}(iR\lambda)}(H_{n}^{(1)}(i\rho\lambda))^{2}=0$$ Here $\alpha\in\mathbb{R}$, $0<R<\rho$, $J_{n}$ is the Bessel function of the first kind of order $n$ and$H_{n}^{(1)}$ is the Hankel function of order $n$.
Moreover $\lambda$ must lie in the region $-\pi<\text{Arg}(\lambda)\le\frac{\pi}{2}$.
Maybe there is a closed form for the sum, but could not find it. I wondered also if I could expand the two linear terms in some Bessel function based series and find a set of recurring equation to solve the problem.
Any insight is appreciated. Thank you.