I've been using the Christoffel-Darboux identity for the Hermite polynomials, $$\sum_{k=0}^n\frac{H_k(x)H_k(y)}{2^k k!}=\frac{1}{2^{n+1} n!}\frac{H_{n+1}(x)H_n(y)-H_n(x)H_{n+1}(y)}{x-y},$$ for some time, and it's been quite helpful. I would like to extend this to a sum of the form $$\sum_{k=0}^n\frac{i^kH_k(x)H_k(y)}{2^k k!}$$ (or if possible an arbitrary phase $e^{ik\theta}$ replacing $i^k$), but I came up empty when looking for references. Can anyone point me in the right direction? or is this a lost cause? Cheers!

EDIT, to provide some motivation: in this paper we explore eigenstates of the position quadrature in a truncated quantum harmonic oscillator number basis. The sum I'm asking about is related to the momentum eigenstates in the truncated space, or equivalently to the Fourier transform of the position eigenstates.