  Properties of certain $q$-orthogonal polynomials are connected to the
$q$-oscillator algebra. The Wall and $q$-Laguerre polynomials are shown to
arise as matrix elements of $q$-exponentials of the generators in a
representation of this algebra. A realization is presented where the continuous
$q$-Hermite polynomials form a basis of the representation space. Various
identities are interpreted within this model. In particular, the connection
formula between the continuous big $q$-Hermite polynomials and the continuous
$q$-Hermite polynomials is thus obtained, and two generating functions for
these last polynomials are algebraically derived.
