We show generic existence of power series a with complex coefficients a_n, such that the sequence of partial sums of a new power series where its coefficients b_n are functions of a_0, a_1, ..., a_n approximate every polynomial uniformly on every compact set K not containing the origin and with connected complement. The functions b_n are assumed to be continuous and such that for every complex numbers a_0, a_1, ... , a_{n - 1}, c there exists a complex number a_n such that b_n(a_0, a_1,..., a_{n-1}, a_n) = c. This clearly covers the case of linear functions b_n.

中文翻译：

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Seleznev 通用动力系列的扩展

我们展示了具有复系数 a_n 的幂级数 a 的一般存在，使得一个新的幂级数的部分和序列，其中其系数 b_n 是 a_0, a_1, ..., a_n 的函数，在每个紧致集合 K 上均匀地逼近每个多项式不包含原点和连接补语。假设函数 b_n 是连续的，并且对于每个复数 a_0, a_1, ... , a_{n - 1}, c 都存在一个复数 a_n 使得 b_n(a_0, a_1,..., a_ {n-1}, a_n) = c。这显然涵盖了线性函数 b_n 的情况。