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We investigate the relations between the rings E , G and D of values taken at algebraic points by arithmetic Gevrey series of order either −1 (E -functions), 0 (analytic continuations of G -functions) or 1 (renormalization of divergent series solutions at ∞ of E -operators) respectively. We prove in particular that any element of G can be written as multivariate polynomial with algebraic coefficients in elements of E and D , and is the limit at infinity of some E -function along some direction. This prompts to defining and studying the notion of mixed functions, which generalizes simultaneously E -functions and arithmetic Gevrey series of order 1. Using natural conjectures for arithmetic Gevrey series of order 1 and mixed functions (which are analogues of a theorem of André and Beukers for E -functions) and the conjecture D ∩ E = Q ‾ (but not necessarily all these conjectures at the same time), we deduce a number of interesting Diophantine results such as an analogue for mixed functions of Beukers' linear independence theorem for values of E -functions, the transcendence of the values of the Gamma function and its derivatives at all non-integral algebraic numbers, the transcendence of Gompertz constant as well as the fact that Euler's constant is not in E. [ABSTRACT FROM AUTHOR]

A classical problem due to Abel is to determine if a differential equation |$y^{\prime}=\eta y$| admits a non-trivial solution |$y$| algebraic over |$\mathbb C(x)$| when |$\eta $| is a given algebraic function over |$\mathbb C(x)$|⁠. Risch designed an algorithm that, given |$\eta $|⁠ , determines whether there exists an algebraic solution or not. In this paper, we adopt a different point of view when |$\eta $| admits a Puiseux expansion with rational coefficients at some point in |$\mathbb C\cup \{\infty \}$|⁠ , which can be assumed to be 0 without loss of generality. We prove the following arithmetic characterization: there exists a non-trivial algebraic solution of |$y^{\prime}=\eta y$| if and only if the coefficients of the Puiseux expansion of |$x\eta (x)$| at |$0$| satisfy Gauss congruences for almost all prime numbers. We then apply our criterion to hypergeometric series: we completely determine the equations |$y^{\prime}=\eta y$| with an algebraic solution when |$x\eta (x)$| is an algebraic hypergeometric series with rational parameters, and this enables us to prove a prediction Golyshev made using the theory of motives. We also present two other applications, namely to diagonals of rational fractions and to directed two-dimensional walks. [ABSTRACT FROM AUTHOR]

Every polynomial P (X) ∈ ℤ [ X ] satisfies the congruences P (n + m) ≡ P (n) mod m for all integers n , m ≥ 0. An integer valued sequence (a n) n ≥ 0 is called a pseudo-polynomial when it satisfies these congruences. Hall characterized pseudo-polynomials and proved that they are not necessarily polynomials. A long-standing conjecture of Ruzsa says that a pseudo-polynomial a n is a polynomial as soon as limsup n | a n | 1 / n < e. Under this growth assumption, Perelli and Zannier proved that the generating series ∑ n = 0 ∞ a n z n is a G -function. A primary pseudo-polynomial is an integer valued sequence (a n) n ≥ 0 such that a n + p ≡ a n mod p for all integers n ≥ 0 and all prime numbers p. The same conjecture has been formulated for them, which implies Ruzsa's, and this paper revolves around this conjecture. We obtain a Hall type characterization of primary pseudo-polynomials. We give a new proof and generalize a result due to Zannier that any primary pseudo-polynomial with an algebraic generating series is a polynomial. We make the Perelli–Zannier Theorem effective and we prove a Pólya type result. [ABSTRACT FROM AUTHOR]

The Hurwitz zeta function ζ (s , a) admits a well-known (divergent) asymptotic expansion in powers of 1 / a involving the Bernoulli numbers. Using Wilson orthogonal polynomials, we determine an effective bound for the error made when this asymptotic series is replaced by nearly diagonal Padé approximants. By specialization, we obtain new fast converging sequences of rational approximations to the values of the Riemann zeta function at every integers ≥2. The latter can be viewed, in a certain sense, as analogues of Apéry's celebrated sequences of rational approximations to ζ (2) and ζ (3). [ABSTRACT FROM AUTHOR]

Given any non-polynomial G-function F(z)=∑∞k=0Akzk of radius of convergence R and in the kernel a G-operator LF, we consider the G-functions F[s]n(z)=∑∞k=0Ak(k+n)szk for every integers s≥0 and n≥1. These functions can be analytically continued to a domain DF star-shaped at 0 and containing the disk {|z|0 and vF>0. This appears to be the first Diophantine result for values of G-functions evaluated outside their disk of convergence. This theorem encompasses a previous result of the authors in [{\em Linear independence of values of G-functions}, 46 pages, J. Europ. Math. Soc., to appear], where α∈Q¯¯¯¯∗ was assumed to be such that |α|

In this paper we define and study a filtration (Gs)s≥0 on the algebra of values at algebraic points of analytic continuations of G-functions: Gs is the set of values at algebraic points in the disk of convergence of all Gfunctions Σ∞n=0 anzn for which there exist some positive integers b and c such that dsbncn+1an is an algebraic integer for any n, where dn = lcm(1, 2, . . ., n). We study the situation at the boundary of the disk of convergence, and using transfer results from analysis of singularities we deduce that constants in Gs appear in the asymptotic expansion of such a sequence (an). [ABSTRACT FROM AUTHOR]

We continue our investigation of E-operators, in particular their connection with Goperators; these differential operators are fundamental in understanding the diophantine properties of Siegel's E and G-functions. We study in detail microsolutions (in Kashiwara's sense) of Fuchsian differential operators, and apply this to the construction of bases of solutions at 0 and ∞ of any E-operator from microsolutions of a G-operator; this provides a constructive proof of a theorem of André. We also focus on the arithmetic nature of connection constants and Stokes constants between different bases of solutions of E-operators. For this, we introduce and study in details an arithmetic (inverse) Laplace transform that enables one to get rid of transcendental numbers inherent to André's original approach. As an application, we define a set of special values of arithmetic Gevrey series, and discuss its conjectural relation with the ring of exponential periods of Kontsevich-Zagier. [ABSTRACT FROM AUTHOR]

Siegel introduced and studied the class of E-functions in 1929. They are power series, solutions of some linear differential equations, whose Taylor coefficients satisfy certain arithmetic and growth conditions. After the work of Siegel and Shidlovskii, and its refinement by Beukers, on the algebraic relations between values of E-functions over Q, it is important to know when the E-functions solutions of a given differential system of order 1 are algebraically dependent or not over Q(z). In this paper, we give the complete classification of the vector solutions of two-dimensional differential systems of order 1 whose components are algebraically dependent E-functions over Q(z). [ABSTRACT FROM AUTHOR]

Allouche and Mendès France [1] have defined the grade of a formal power series with algebraic coefficients as the smallest integer k such that this series is the Hadamard product of k algebraic power series. In this paper, we obtain lower and upper bounds for the grade of hypergeometric series by comparing two different asymptotic expansions of their Taylor coefficients, one obtained from their definition and another one obtained when assuming that the grade has a certain value. In such expansions, Gamma values at rational points naturally appear and our results mostly depend on the Rohrlich–Lang Conjecture for polynomial relations in Gamma values. We also obtain unconditional and sharp results when we can apply Diophantine results such as the Wolfart–Wüstholz Theorem for Beta values. [ABSTRACT FROM AUTHOR]

We consider a multi-parameter family of canonical coordinates and mirror maps originally introduced by Zudilin. This family includes many of the known one-variable mirror maps as special cases, in particular many of modular origin and the celebrated ‘quintic’ example of Candelas, de la Ossa, Green and Parkes. In a previous paper, we proved that all coefficients in the Taylor expansions at 0 of these canonical coordinates (and, hence, of the corresponding mirror maps) are integers. Here we prove that all coefficients in their Taylor expansions at 0 are positive. Furthermore, we provide several results about the behaviour of the canonical coordinates and mirror maps as complex functions. In particular, we address their analytic continuation, points of singularity, and radius of covergence. We present several very precise conjectures on the radius of covergence of the mirror maps and the sign pattern of the coefficients in their Taylor expansions at 0. [ABSTRACT FROM AUTHOR]

In this paper, we present and solve some very general new Pade´ approximant problems, whose solutions can be expressed with hypergeometric series. These series appear in the proofs of the irrationality of ζ(3), of infinitely many ζ(2n+1), and in essentially all results of this kind in the literature. We also prove two new Diophantine results with this method. [Copyright &y& Elsevier]