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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.

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  • $\begingroup$ 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. $\endgroup$
    – Henry Cohn
    Commented Jun 24, 2012 at 13:49
  • $\begingroup$ The stated identity is out by a factor of 1/2 on the RHS. $\endgroup$
    – Peter Forrester
    Commented Nov 4, 2019 at 3:38
  • $\begingroup$ I initially thought this might be a confusion between the "physicists' convention" and the "probabilists' convention" for the Hermite polynomials (cf., e.g., the wikipedia page on Hermite polynomials), but upon checking the linked paper, the authors are indeed using the "physicists' convention", for which an extra factor 1/2 on the RHS is needed. I've submitted an edit to the OP. $\endgroup$
    – Michael Engelhardt
    Commented Nov 4, 2019 at 5:02

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