Algebraic properties of Э-functions
type: Item
instance of: scholarly article
title: Algebraic properties of Э-functions
zbMATH Open document ID: 1489.11106
author: Timothy Ferguson
published in: Journal of Number Theory
publication date: 2021-09-06T00:00:00Z
review text: An \(E\)-function is a power series \(f(z):=\sum_{n=0}^\infty a_n/n!z^n \in \overline{\mathbb Q}z\) solution of a linear differential equation with coefficients in \(\overline{\mathbb Q}(z)\) and such that the sequence \((a_n)_{n\ge 0}\) satisfies certain Archimedean and non-Archimedean growth conditions. They generalize the exponential function, and were first defined and studied by Siegel in 1929. The celebrated Siegel-Shidlovskii theorem [\textit{A. B. Shidlovskiĭ}, Dokl. Akad. Nauk SSSR 105, 35--37 (1955; Zbl 0065.28304)] states the following: given a vector of \(E\)-functions \(Y:={}^t(f_1, \ldots, f_n)\) solution of a differential system \(Y'=AY\) where \(A\) is a matrix with elements in \(\overline{\mathbb Q}(z)\), the transcendence degree over \(\overline{\mathbb Q}(z)\) of the field generated by \(f_1(z), \ldots, f_n(z)\) over \(\overline{\mathbb Q}(z)\) is equal to the transcendence degree over \(\overline{\mathbb Q}\) of the field generated by \(f_1(\alpha), \ldots, f_n(\alpha)\) over \(\overline{\mathbb Q}\) for any given \(\alpha \in \overline{\mathbb Q}^*\) which is not a pole of some element of \(A\). \textit{Y. André} gave a new proof of this theorem [Ann. Math. (2) 151, No. 2, 741--756 (2000; Zbl 1037.11050)], based in particular on the fact that if \(f(z)\in \mathbb Qz\) is an \(E\)-function such that \(f(1)=0\), then \(f(z)/(z-1)\) is still an \(E\)-function. In this interesting paper, the author considers the class, say \(S\) in this review, of power series \(u(z):=\sum_{n=0}^\infty a_n n!z^n \in \overline{\mathbb Q}z\) solution of a linear differential equation with coefficients in \(\overline{\mathbb Q}(z)\) and such that \(\sum_{n=0}^\infty a_n/n!z^n\) is an \(E\)-function, which were also considered by André. Unless it reduces to a polynomial, a series \(u(z) \in S\) converges only for \(z=0\) but it can be summed to a function \(U(z)\) analytic in ``large sectors'' using Ramis 1-summation process. In particular, if the underlying…
reviewed by: T. Rivoal
Mathematics Subject Classification ID: 11J91
Mathematics Subject Classification ID: 34G10
Mathematics Subject Classification ID: 33F10
zbMATH DE Number: 7391370
zbMATH Keywords: transcendental numbers
zbMATH Keywords: differential equations
zbMATH Keywords: arithmetic special functions
Wikidata QID: Q114157006
MaRDI profile type: Publication
full work available at URL: j.jnt.2021.04.010
OpenAlex ID: W3167141111
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DOI: 10.1016/J.JNT.2021.04.010
description: scientific article; zbMATH DE number 7391370