  We provide a general result for finding Stein operators for the product of
two independent random variables whose Stein operators satisfy a certain
assumption, extending a recent result of Gaunt, Mijoule and Swan \cite{gms18}.
This framework applies to non-centered normal and non-centered gamma random
variables, as well as a general sub-family of the variance-gamma distributions.
Curiously, there is an increase in complexity in the Stein operators for
products of independent normals as one moves, for example, from centered to
non-centered normals. As applications, we give a simple derivation of the
characteristic function of the product of independent normals, and provide
insight into why the probability density function of this distribution is much
more complicated in the non-centered case than the centered case.
