Propagators of isochronous an-harmonic oscillators and Mehler formula for the x-Hermite polynomials

It is shown that fundamental solutions $K^\sigma(x,y;t)=\langle x|e^{-i H^\sigma t}|y\rangle$ of the non-stationary Schr\"{o}dinger equation (Green functions, or propagators) for the rational extensions of the Harmonic oscillator $H^\sigma=H_{\rm osc}+\Delta V^\sigma$ are expressed in terms of elementary functions only. An algorithm to calculate explicitly $K_\sigma$ for an arbitrary $\sigma\in \mathbb{N}$ is given, compact expressions for $K^{\{1,2\}}$ and $K^{\{2,3\}}$ are presented. A generalization of the Mehler's formula to the case of exceptional Hermite polynomials is given.