Hi! I've been using the Christoffel-Darboux identity for the Hermite polynomials, $$\sum_{k=0}^n\frac{H_k(x)H_k(y)}{2^k k!}=\frac{1}{2^n n!}\frac{H_{n+1}(x)H_n(y)-H_n(x)H_{n+1}(y)}{x-y},$$ 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!
Does the sum with $i^k$ have a conceptual meaning (the way the original sum gives a reproducing kernel)? You can certainly compute a Hermite expansion for the sum times $x-y$, but unlike the Christoffel-Darboux case you don't end up with just a few terms, and it's not clear that it's really simpler than what you started with. On the other hand, I can't rule out the possibility of a beautiful formula of another sort. –  Henry Cohn Jun 24 '12 at 13:49