Des crit\`eres de transcendance inspir\'es par un texte de Kolberg dat\'e de 1962

TL;DR

This paper explores transcendence criteria inspired by Kolberg's 1962 work, providing explicit conditions, clarifying proof details, and extending the scope of transcendence results for sums of specific power series at algebraic points.

Contribution

It introduces new transcendence criteria based on Kolberg's theorem, clarifies complex proof aspects, and extends the applicability of these criteria to broader classes of functions.

Findings

Explicit transcendence criteria for power series sums at algebraic points

Clarification of delicate proof points in Kolberg's theorem

Extension of transcendence results to wider function classes

Abstract

Transcendence criteria inspired by Kolberg's paper dated 1962. In his paper dated 1962, Kolberg states and proves a theorem on the transcendence of the values of the sums of a class of certain power series in x, for algebraic values of x. It builds on Lindemann's theorem and uses, en passant, a transcendence criterion for the values of certain rational fonctions. This last criterion is made explicit. We clarify certain delicate points of the proof and show how to extend its scope.