We establish generic existence of Universal Taylor Series on products \(\varOmega = \prod \varOmega _i\) of planar simply connected domains \(\varOmega _i\) where the universal approximation holds on products K of planar compact sets with connected complements provided \(K \cap \varOmega = \emptyset \). These classes are with respect to one or several centers of expansion and the universal approximation is at the level of functions or at the level of all derivatives. Also, the universal functions can be smooth up to the boundary, provided that \(K \cap \overline{\varOmega } = \emptyset \) and \(\{\infty \} \cup [{\mathbb {C}} {\setminus } \overline{\varOmega }_i]\) is connected for all i. All previous kinds of universal series may depend on some parameters; then the approximable functions may depend on the same parameters, as it is shown in the present paper.

我们在平面简单连接域\（\ varOmega _i \）的乘积\（\ varOmega = \ prod \ varOmega _i \）上建立通用泰勒级数的泛型存在，其中通用近似成立于具有连接补码的平面紧集的乘积K \（K \ cap \ varOmega = \ emptyset \）。这些类别是关于一个或几个扩展中心的，并且通用近似是在功能级别或在所有导数级别。另外，只要\（K \ cap \ overline {\ varOmega} = \ emptyset \）和\（\ {\ infty \} \ cup [{\ mathbb {C}} {\ setminus} \ overline {\ varOmega} _i] \）连接了我所有的人。先前所有的通用系列可能取决于某些参数。那么近似函数可能取决于相同的参数，如本文所示。