205 research outputs found
On the denominators of the Taylor coefficients of G-functions
Let be a
-function, and, for any , let denote the least
integer such that are all
algebraic integers. By definition of a -function, there exists some constant
such that for all . In practice, it is
observed that always divides where
, are positive integers and is an
integer. We prove that this observation holds for any -function provided the
following conjecture is assumed: {\em Let be a number field, and
be a -operator; then the generic radius
of solvability is equal to 1, for all finite places of
except a finite number.} The proof makes use of very precise
estimates in the theory of -adic differential equations, in particular the
Christol-Dwork Theorem. Our result becomes unconditional when is a
geometric differential operator, a special type of -operators for which the
conjecture is known to be true. The famous Bombieri-Dwork Conjecture asserts
that any -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
admits a non-trivial solution algebraic over
when is a given algebraic function over . Risch designed
an algorithm that, given , determines whether there exists an algebraic
solution or not. In this paper, we adopt a different point of view when
admits a Puiseux expansion with rational coefficients at some point in , 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 if and only if the coefficients of the
Puiseux expansion of at satisfy Gauss congruences for almost all
prime numbers. We then apply our criterion to hypergeometric series: we
completely determine the equations with an algebraic solution when
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, 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 and closed-form expressions for a class of zeta series
In a recent work, Dancs and He found an Euler-type formula for
, 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 , which is a rational multiple
of . For the Dirichlet beta function, the things are `inverse':
is a rational multiple of and no closed-form
expression is known for . Here in this work, I modify the Dancs-He
approach in order to derive an Euler-type formula for ,
including , 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 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
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
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