  We discuss the general matrix elements of the squeezing operator between
number eigenstates of a harmonic oscillator (which may also represent a
quantized mode of the electromagnetic radiation). These matrix elements have
first been used by Popov and Perelomov (1969) long ago, in their thorough
analysis of the parametric excitation of harmonic oscillators. They expressed
the matrix elements in terms of transcendental functions, the associated
Legendre functions. In the present paper we will show that these matrix
elements can also be expressed by the classical Gegenbauer polynomials. This
new expression makes it possible to determine coherent and incoherent
superpositions of these matrix elements in closed analytic forms. As an
application, we describe multiphoton transitions in the system "charged
particle + electromagnetic radiation", induced by a (strong) coherent field or
by a black-body radiation component (with a Planck-Bose photon number
distribution). The exact results are compared with the semi-classical ones. We
will show that in case of interaction with a thermal field, the semi-classical
result (with a Gaussian stochastic field amplitude) yields an acceptable
approximation only in the Rayleigh-Jeans limit, however, in the Wien limit it
completely fails.
