Linear finite difference operators preserving Laguerre-P\'olya class

We completely describe all finite difference operators of the form $${\mathrm{\Delta }}_{M_1,M_2,h}(f)(z)=M_1(z)f(z+h)+M_2(z)f(z-h)$$ preserving the Laguerre-P'olya class of entire functions. Here $M_1$ and $M_2$ are some complex functions and $h$ is a nonzero complex number.