# An extension of the universal power series of Seleznev

## TL;DR

This paper proves the generic existence of power series with complex coefficients that can approximate any polynomial uniformly on certain compact sets, extending Seleznev's universal power series concept.

## Contribution

It introduces a broad class of power series with coefficients depending on previous terms, generalizing previous universal approximation results.

## Key findings

- Power series can approximate any polynomial uniformly on specified compact sets.
- The constructed power series have coefficients that depend continuously on previous coefficients.
- The results include the case where the coefficients are linear functions of previous coefficients.

## Abstract

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.