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I was wondering if someone could tell me when the following relation holds? where $H_{n}(x)$ are Hermite polynomials and $\delta(x-x')$ is Dirac delta function: $$ \sum_{n=0}^\infty \frac{1}{\sqrt{\pi}2^{n}n!} e^{-\frac{1}{2}x^{2}-\frac{1}{2}x'^{2}} H_{n}(x)H_{n}(x') = \delta(x-x') . $$

I asked this because I am trying to solve this:

https://physics.stackexchange.com/questions/455890/photon-number-representation-of-a-position-eigenstate

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Denote the left hand side of your identity by $K(x,x’)$. The identity $K(x,x’)=\delta(x-x’)$ is an identity between generalized functions. What it’s really saying is that $$ \int_{-\infty}^{\infty} K(x,x’)f(x’)dx’ = f(x) $$ for any $x$ and any reasonably nice function $f$. In fact it’s enough to prove this for $f(x)=e^{-x^2/2} H_m(x)$ a Hermite function, since the Hermite functions are s complete orthogonal system in $L_2(\mathbb{R})$. For this $f$ the claim follows immediately from the orthogonality relation for the Hermite polynomials, since when we perform the integration for the $n$th term in the sum, we get 0 if $n \neq m$ and the correct thing for $n=m$.

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