= Mehler formula for Hermite functions
{c}
{title2=$\sum_{n\geq0}t^nH_n(x)H_n(y)=\frac{\exp\!\left(\frac{4xyt-(1+t^2)(x^2+y^2)}{2(1-t^2)}\right)}{\sqrt{\pi(1-t^2)}}$}
For $|t|<1$, this identity sums the <Hermite function> expansion into a Gaussian. One proof takes the <Bargmann-Fock space> inner product of the generating functions $\pi^{-1/4}\exp(-x^2/2+\sqrt2 xz-z^2/2)$ at $\sqrt t\,z$. Writing $z=u+iv$, the two resulting real <Gaussian integrals> have quadratic coefficients $1+t$ and $1-t$ and linear coefficients $\sqrt{2t}(x+y)$ and $i\sqrt{2t}(x-y)$. Evaluation gives the formula. Setting $t=e^{-2s}$ and multiplying by $e^{-s}$ gives the <harmonic oscillator transition kernel> of $e^{-s(-d^2/dx^2+x^2)}$.
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