Séries Gevrey de type arithmétique, II. Transcendance sans transcendanceReport as inadecuate



 Séries Gevrey de type arithmétique, II. Transcendance sans transcendance


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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,







Author: Yves André

Source: https://archive.org/



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