  The expansion of quantum states and operators in terms of Fock states plays a
fundamental role in the field of continuous-variable quantum mechanics. In
particular, for general single-mode Gaussian operators and Gaussian noisy
states, many different approaches have been used in the evaluation of their
Fock representation. In this paper a natural approach has been applied using
exclusively the operational properties of the Hermite and Hermite-like
polynomials and showing their fundamental role in this field. Closed-form
results in terms of polynomials, exponentials, and simple algebraic functions
are the major contribution of the paper.
