S\'eries Gevrey de type arithm\'etique, II. Transcendance sans transcendance

Abstract

In this second part, we study the Diophantine properties of values of arithmetic Gevrey series of non-zero order at algebraic points. We rely on the fact, proved in the first part, that the minimal differential operator (with polynomial coefficients) which annihilates such a series has no non-trivial singularity outside the origin and infinity. We show how to draw from this fact some transcendence properties, and recover in particular the fundamental theorem of the Siegel-Shidlovsky theory on algebraic independence of values of E-functions. The paradox of the title points out the contrast between the qualitative aspect of this new argument and the essentially quantitative aspect of the traditional approach. At last, we discuss q-analogues of the theory (theta-functions, q-exponential,...).