Maronikolakis, K.; Nestoridis, V. An extension of the universal power series of Seleznev. (English) Zbl 1451.30114 Result. Math. 75, No. 4, Paper No. 131, 8 p. (2020). In this very interesting paper the authors expand a classical result due to A. I. Seleznev [Mat. Sb., N. Ser. 28(70), 453–460 (1951; Zbl 0043.29501)]. In particular they prove the generic existence of power series \(\sum_{n=0}^{+\infty}a_nz^n\), such that the partial sums \(\sum_{n=0}^{N}b_n(a_0,a_1,\ldots,a_n)z^n, \ N\in\mathbb{N}\), approximate all polynomials on every compact subset in \(\mathbb{C}\setminus\{0\}\) with connected complement. The functions \(b_n: \mathbb{C}^{n+1}\to\mathbb{C}\) are assumed to be continuous and the functions \(b_n(a_0,a_1,\ldots, a_{n-1}, \cdot):\mathbb{C}\to \mathbb{C}\) are assumed to be onto \(\mathbb{C}\) for every choice of \(a_0,a_1\ldots,a_{n-1}\in\mathbb{C}\). In order to prove the result, the authors follow methods used in Universal Taylor Series. Seleznev’s result corresponds with case \(b_n(a_0,a_1,\ldots,a_n)=a_n, \ n\in\mathbb{N}\). Another interesting result, connected with Cesàro summability, is the case where \(b_n(a_0,a_1,\ldots,a_n)=\dfrac{a_0+a_1+\ldots+a_n}{n+1}, \ n\in\mathbb{N}.\) Reviewer: Vagia Vlachou (Patras) Cited in 1 Document MSC: 30K05 Universal Taylor series in one complex variable 30E10 Approximation in the complex plane Keywords:universal Taylor series; Mergelyan theorem Citations:Zbl 0043.29501 PDFBibTeX XMLCite \textit{K. Maronikolakis} and \textit{V. Nestoridis}, Result. Math. 75, No. 4, Paper No. 131, 8 p. (2020; Zbl 1451.30114) Full Text: DOI arXiv References: [1] Bayart, F.; Grosse-Erdmann, K-G; Nestoridis, V.; Papadimitropoulos, C., Abstract theory of universal series and applications, Proc. London Math. Soc., 3, 96, 417-463 (2008) · Zbl 1147.30003 [2] Chui, C.; Parnes, MN, Approximation by overconvergence of power series, J. Math. Anal. Appl., 36, 693-696 (1971) · Zbl 0224.30006 [3] Luh, W., Approximation analytischer Funktionen durch überkonvergente Potenzreihen und deren Matrix-Transformierten, Mitt. Math. Sem. Giessen, 88, 1-56 (1970) · Zbl 0231.30005 [4] Nestoridis, V., Universal Taylor series, Ann. Inst. Fourier, 46, 1293-1306 (1996) · Zbl 0865.30001 [5] Seleznev, AI, On universal power series’ (Russian), Mat. Sbornik (N. S.), 28, 453-460 (1951) [6] Vlachou, V., On some classes of universal functions, Analysis, 22, 149-161 (2002) · Zbl 1001.30004 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.