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Arithmetic theory of $$E$$-operators. (Théorie arithmétique des $$E$$-opérateurs.) (English. French summary) Zbl 1370.11090
In a previous paper [Comment. Math. Helv. 89, No. 2, 313–341 (2014; Zbl 1304.11070)], the authors introduced and studied the subring $${\mathbb{G}}$$ of $${\mathbb{C}}$$ which is the set of values at algebraic points of analytic continuations of $$G$$-functions. They suggested that $${\mathbb{G}}$$ may be $${\mathcal{P}}[1/\pi]$$, where $${\mathcal{P}}$$ is the ring of periods as defined by M. Kontsevich and D. Zagier [in: Mathematics unlimited – 2001 and beyond. Berlin: Springer. 771–808 (2001; Zbl 1039.11002)].
Here, they introduce and study the subring $${\mathbb{E}}$$ of $${\mathbb{C}}$$ which is the set of values taken at any algebraic point by any $$E$$-function. They suggest that $${\mathbb{E}}$$ may be contained in the ring generated by $$1/\pi$$ and exponential periods.
An $$E$$-function is a solution of a differential $$E$$-operator $$L$$, with a regular singularity at $$0$$ and an irregular singularity at $$\infty$$. At a regular point $$\alpha\in{\overline{\mathbb{Q}}}\setminus\{0\}$$, there is a basis of holomorphic solutions in $${\overline{\mathbb{Q}}}[[z-\alpha]]$$; around $$z=\alpha$$, a solution $$F$$ of $$L$$ at $$z=0$$ is a linear combination of the elements of this basis, the coefficients of which are called the connection constants. From a result of Y. André [Ann. Math. (2) 151, No. 2, 705–740 (2000; Zbl 1037.11049)], it follows that $$F$$ is a linear combination of $$E$$-functions, powers of $$z$$ and powers of $$\log z$$. Assuming that the coefficients of this linear combination are algebraic, the authors prove that the connection constants belong to $${\mathbb{E}}[\log\alpha]$$. When $$F$$ is an $$E$$-function, they belong to $${\mathbb{E}}$$.
The authors also study the local solutions at infinity, which involve divergent series. The asymptotic expansions are valid in large sectors $$\arg(z)\in [\theta-(\pi/2)-\varepsilon,\theta+(\pi/2)+\varepsilon]$$ and involve functions $$e^{\varrho z}$$, $$z^{-n-\alpha}$$ and powers of $$\log (1/z)$$. For an $$E$$-operator, André [loc. cit.] has constructed a basis of formal solutions at infinity involving divergent Gevrey series of order $$1$$. Given an $$E$$-function $$F$$ and a direction $$\theta$$ which is not an anti Stoke constant, the coefficients of $$F$$ in this basis are called the Stoke constants. The authors prove that they belong to the ring $${\mathbb{S}}$$ that they define as the $${\mathbb{G}}$$-submodule of $${\mathbb{C}}$$ generated by all values of the derivatives of the Gamma function $$\Gamma(z)$$ at algebraic points. They show that $${\mathbb{S}}$$ is also the $${\mathbb{G}}[\gamma]$$-module generated by all values of the derivatives of the Gamma function $$\Gamma(z)$$ at algebraic points, where $$\gamma$$ is Euler’s constant.
In their previous paper on $$G$$-functions [loc. cit.], the authors introduced the definition of $$G$$-approximation. Here, they introduce a similar notion which is $$E$$-approximation, and they study the set of numbers that have $$E$$-approximations.

##### MSC:
 11J91 Transcendence theory of other special functions 33E30 Other functions coming from differential, difference and integral equations 34M40 Stokes phenomena and connection problems (linear and nonlinear) for ordinary differential equations in the complex domain 44A10 Laplace transform
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