  We build upon recent advances on the distributional aspect of Stein's method
to propose a novel and flexible technique for computing Stein operators for
random variables that can be written as products of independent random
variables. We show that our results are valid for a wide class of distributions
including normal, beta, variance-gamma, generalized gamma and many more. Our
operators are $k$th degree differential operators with polynomial coefficients;
they are straightforward to obtain even when the target density bears no
explicit handle. As an application, we derive a new formula for the density of
the product of $k$ independent symmetric variance-gamma distributed random
variables.
