A version of Fabry's theorem for power series with regularly varying coefficients

TL;DR

This paper extends Fabry's theorem to power series with coefficients that have a regular variation property, linking the sign change frequency to the series' analytic continuation.

Contribution

It introduces a stronger version of Fabry's theorem applicable to power series with regularly varying coefficients, enhancing understanding of their analytic properties.

Findings

Established a new relation between sign change frequency and analytic continuation

Extended Fabry's theorem to a broader class of power series

Provided conditions for the analytic continuation based on coefficient behavior

Abstract

For real power series whose non-zero coefficients satisfy $|a_m|^{1/m}\to~1$ we prove a stronger version of Fabry theorem relating the frequency of sign changes in the coefficients and analytic continuation of the sum of the power series.