  In the context of phase-space quantization, matrix elements and observables
result from integration of c-number functions over phase space, with Wigner
functions serving as the quasi-probability measure. The complete sets of Wigner
functions necessary to expand all phase-space functions include off-diagonal
Wigner functions, which may appear technically involved. Nevertheless, it is
shown here that suitable generating functions of these complete sets can often
be constructed, which are relatively simple, and lead to compact evaluations of
matrix elements. New features of such generating functions are detailed and
explored for integer-indexed sets, such as for the harmonic oscillator, as well
as continuously indexed ones, such as for the linear potential and the
Liouville potential. The utility of such generating functions is illustrated in
the computation of star functions, spectra, and perturbation theory in phase
space.
