Mehler's formula

Mehler (1866) defined a function

${\displaystyle E(x,y)={\frac {1}{\sqrt {1-\rho ^{2}}}}\exp \left(-{\frac {\rho ^{2}(x^{2}+y^{2})-2\rho xy}{(1-\rho ^{2})}}\right)~,}$

and showed, in modernized notation, that it can be expanded in terms of Hermite polynomials H(.) based on weight function exp(−x²) as

This result is useful, in modified form, in quantum physics, probability theory, and harmonic analysis.

In physics, the fundamental solution, (Green's function), or propagator of the Hamiltonian for the quantum harmonic oscillator is called the Mehler kernel. It provides the fundamental solution---the most general solutionφ(x,t) to

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