Algebraic properties of Э-functions
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 \(G\)-function \(g(z):=\sum_{n=0}^\infty a_nz^n\) is analytic in a sector \(\{z \in \mathbb C : \vert \arg(z)\vert <\varepsilon\}\) for some \(\varepsilon>0\), then so is \(U(z)\). Under this analytic assumption, the author formulates the following natural conjecture: given \(u(z)\in S\), \(u(z)/(z-1)\) is in \(S\) if and only if \(U(1)=0\). In the main theorem of the paper and in the same context as above, the author formulates eight properties that he proves to be all equivalent to the fact that \(u(z)/(z-1)\) is in \(S\), many of them being formulated in term of the differential equations satisfied by \(g(z)\) and \(\int_0^z e^{z-x}g(x)dx\). He then uses these properties to prove that, under his conjecture above, the numbers 1, \(\int_0^{+\infty} e^{-x}dx/(x+\alpha_1), \dots, \int_0^{+\infty} e^{-x}dx/(x+\alpha_n)\) are linearly independent over \(\mathbb Q\) for any distinct positive rational numbers \(\alpha_1, \ldots, \alpha_n\), \(n\ge 1\). That conditional Diophantine result includes the irrationality of Gompertz constant \(\int_0^{+\infty} e^{-x}dx/(x+1)\). This paper can be viewed as a first step towards an analogue of the Siegel-Shidlovskii theorem for the series in \(S\).
scientific article; zbMATH DE number 7391370