New asymptotic formulae for the two-dimensional Hermite polynomials with large values of indices are found. The applications to the photon distribution functions in squeezed one-mode mixed quantum states are considered.
The photon distribution function of an N-mode mixed state of light described by the Wigner function of generic Gaussian form is calculated explicitly in terms of the Hermite polynomials of 2N variables with equal pairs of indices. Simple formulas for the mean values and dispersions of photon numbers are found. The N-mode photon distribution functions for squeezed photon number states and for squeezed coherent states are expressed in terms of the Hermite polynomials of 2N and N variables, respectively.