On the denominators of the Taylor coefficients of G-functions

Let $\sum\_{n=0}^\infty a\_n z^n\in \overline{\mathbb Q}[[z]]$ be a $G$-function, and, for any $n\ge0$, let $\delta\_n\ge 1$ denote the least integer such that $\delta\_n a\_0, \delta\_n a\_1, ..., \delta\_n a\_n$ are all algebraic integers. By definition of a $G$-function, there exists some constant $c\ge 1$ such that $\delta\_n\le c^{n+1}$ for all $n\ge 0$. In practice, it is observed that $\delta\_n$ always divides $D\_{bn}^{s} C^{n+1}$ where $D\_n=lcm\{1,2, ..., n\}$, $b, C$ are positive integers and $s\ge 0$ is an integer. We prove that this observation holds for any $G$-function provided the following conjecture is assumed: {\em Let $\mathbb{K}$ be a number field, and $L\in \mathbb{K}[z,\frac{d }{d z}]$ be a $G$-operator; then the generic radius of solvability $R\_v(L)$ is equal to 1, for all finite places $v$ of $\mathbb{K}$ except a finite number.} The proof makes use of very precise estimates in the theory of $p$-adic differential equations, in particular the Christol-Dwork Theorem. Our result becomes unconditional when $L$ is a geometric differential operator, a special type of $G$-operators for which the conjecture is known to be true. The famous Bombieri-Dwork Conjecture asserts that any $G$-operator is of geometric type, hence it implies the above conjecture

On Abel's problem and Gauss congruences

A classical problem due to Abel is to determine if a differential equation $y'=\eta y$ 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 $y'=\eta y$ if and only if the coefficients of the Puiseux expansion of $\eta$ at $0$ satisfy Gauss congruences for almost all prime numbers. We then apply our criterion to hypergeometric series: we completely determine the equations $y'=\eta y$ 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 three other applications, in particular to diagonals of rational fractions and to directed two-dimensional walks

On Siegel's problem for E-functions

In this new version, a similar problem for G-functions is considered in Section 6.Siegel defined in 1929 two classes of power series, the E-functions and G-functions, which generalize the Diophantine properties of the exponential and logarithmic functions respectively. In 1949, he asked whether any E-function can be represented as a polynomial with algebraic coefficients in a finite number of confluent hypergeometric series with rational parameters. The case of E-functions of differential order less than 2 was settled in the affirmative by Gorelov in 2004, but Siegel's question is open for higher order. We prove here that if Siegel's question has a positive answer, then the ring G of values taken by analytic continuations of G-functions at algebraic points must be a subring of the relatively "small" ring H generated by algebraic numbers, $1/\pi$ and the values of the derivatives of the Gamma function at rational points. Because that inclusion seems unlikely (and contradicts standard conjectures), this points towards a negative answer to Siegel's question in general. As intermediate steps, we first prove that any element of G is a coefficient of the asymptotic expansion of a suitable E-function, which completes previous results of ours. We then prove that the coefficients of the asymptotic expansion of a confluent hypergeometric series with rational parameters are in H. Finally, we prove a similar result for G-functions

An Euler-type formula for β(2n)\beta(2n) and closed-form expressions for a class of zeta series

In a recent work, Dancs and He found an Euler-type formula for $\,\zeta{(2\,n+1)}$, $\,n\,$ being a positive integer, which contains a series they could not reduce to a finite closed-form. This open problem reveals a greater complexity in comparison to $\zeta(2n)$, which is a rational multiple of $\pi^{2n}$. For the Dirichlet beta function, the things are `inverse': $\beta(2n+1)$ is a rational multiple of $\pi^{2n+1}$ and no closed-form expression is known for $\beta(2n)$. Here in this work, I modify the Dancs-He approach in order to derive an Euler-type formula for $\,\beta{(2n)}$, including $\,\beta{(2)} = G$, the Catalan's constant. I also convert the resulting series into zeta series, which yields new exact closed-form expressions for a class of zeta series involving $\,\beta{(2n)}$ and a finite number of odd zeta values. A closed-form expression for a certain zeta series is also conjectured.Comment: 11 pages, no figures. A few small corrections. ACCEPTED for publication in: Integral Transf. Special Functions (09/11/2011

Séries hypergéométriques multiples et polyzêtas

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Implications of Cosmological Gamma-Ray Absorption II. Modification of gamma-ray spectra

Bearing on the model for the time-dependent metagalactic radiation field developed in the first paper of this series, we compute the gamma-ray attenuation due to pair production in photon-photon scattering. Emphasis is on the effects of varying the star formation rate and the fraction of UV radiation assumed to escape from the star forming regions, the latter being important mainly for high-redshift sources. Conversely, we investigate how the metagalactic radiation field can be measured from the gamma-ray pair creation cutoff as a function of redshift, the Fazio-Stecker relation. For three observed TeV-blazars (Mkn501, Mkn421, H1426+428) we study the effects of gamma-ray attenuation on their spectra in detail.Comment: 10 pages, 6 figures, accepted by A&

Spectrum and Variability of Mrk501 as observed by the CAT Imaging Telescope

The CAT Imaging Telescope has observed the BL Lac object Markarian 501 between March and August 1997. We report here on the variability over this time including several large flares. We present also preliminary spectra for all these data, for the low emission state, and for the largest flare.Comment: 4 pages, 4 figures, Late