La s\'erie enti\`ere $1+\frac z{\Gamma(1+i)}+\frac{z^2}{\Gamma(1+2i)}+\frac{z^3}{\Gamma(1+3i)}+...$ poss\`ede une fronti\`ere naturelle~!

TL;DR

This paper investigates a family of non-lacunary power series with coefficients involving the Gamma function, demonstrating they have a natural boundary by transforming them into lacunary Dirichlet series, revealing complex behavior of the Gamma function.

Contribution

It introduces a method to transform certain Gamma coefficient series into lacunary Dirichlet series, establishing the existence of natural boundaries for these series.

Findings

Series with Gamma function coefficients have natural boundaries.

Transformation into lacunary Dirichlet series is possible.

Gamma function exhibits unpredictable behavior on vertical lines.

Abstract

The lacunary series are the most classical examples among all the power series whose circle of convergence constitutes a natural boundary (\cite[\S~93-94, p.~372-383]{Di}, \cite[\S 7.43, p.~223]{Ti}, ...). In this Note, we study a family of non-lacunary power series whose coefficients are given by means of values of the Gamma function over vertical line. We explain how to transform these series into lacunary Dirichlet series, which allows us to conclude the existence of their natural boundary. Our results, which illustrate in what manner the Gamma function may have a unpredictable behaviour on any vertical line, may also be partially understood in the framwork of our forthcoming work on a class of differential $q$-difference equations, namely, on pantagraph type equations (see \cite{KM} for instance).