Universal power series of Seleznev with parameters in several variables

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