  We generalize the universal power series of Seleznev to several variables and
we allow the coefficients to depend on parameters. Then, the approximable
functions may depend on the same parameters. The universal approximation holds
on products $K = \displaystyle\prod_{i = 1}^d K_i$, where $K_i \subseteq
\mathbb{C}$ are compact sets and $\mathbb{C} \setminus K_i$ are connected, $i =
1, \dots, d$ and $0 \notin K$. On such $K$ the partial sums approximate
uniformly any polynomial. Finally, the partial sums may be replaced by more
general expressions. The phenomenon is topologically and algebraically generic.
