433 research outputs found
Denominator Bounds and Polynomial Solutions for Systems of q-Recurrences over K(t) for Constant K
We consider systems A_\ell(t) y(q^\ell t) + ... + A_0(t) y(t) = b(t) of
higher order q-recurrence equations with rational coefficients. We extend a
method for finding a bound on the maximal power of t in the denominator of
arbitrary rational solutions y(t) as well as a method for bounding the degree
of polynomial solutions from the scalar case to the systems case. The approach
is direct and does not rely on uncoupling or reduction to a first order system.
Unlike in the scalar case this usually requires an initial transformation of
the system.Comment: 8 page
Extended Rate, more GFUN
We present a software package that guesses formulae for sequences of, for
example, rational numbers or rational functions, given the first few terms. We
implement an algorithm due to Bernhard Beckermann and George Labahn, together
with some enhancements to render our package efficient. Thus we extend and
complement Christian Krattenthaler's program Rate, the parts concerned with
guessing of Bruno Salvy and Paul Zimmermann's GFUN, the univariate case of
Manuel Kauers' Guess.m and Manuel Kauers' and Christoph Koutschan's
qGeneratingFunctions.m.Comment: 26 page
NumGfun: a Package for Numerical and Analytic Computation with D-finite Functions
This article describes the implementation in the software package NumGfun of
classical algorithms that operate on solutions of linear differential equations
or recurrence relations with polynomial coefficients, including what seems to
be the first general implementation of the fast high-precision numerical
evaluation algorithms of Chudnovsky & Chudnovsky. In some cases, our
descriptions contain improvements over existing algorithms. We also provide
references to relevant ideas not currently used in NumGfun
Symbolic-Numeric Tools for Analytic Combinatorics in Several Variables
Analytic combinatorics studies the asymptotic behaviour of sequences through
the analytic properties of their generating functions. This article provides
effective algorithms required for the study of analytic combinatorics in
several variables, together with their complexity analyses. Given a
multivariate rational function we show how to compute its smooth isolated
critical points, with respect to a polynomial map encoding asymptotic
behaviour, in complexity singly exponential in the degree of its denominator.
We introduce a numerical Kronecker representation for solutions of polynomial
systems with rational coefficients and show that it can be used to decide
several properties (0 coordinate, equal coordinates, sign conditions for real
solutions, and vanishing of a polynomial) in good bit complexity. Among the
critical points, those that are minimal---a property governed by inequalities
on the moduli of the coordinates---typically determine the dominant asymptotics
of the diagonal coefficient sequence. When the Taylor expansion at the origin
has all non-negative coefficients (known as the `combinatorial case') and under
regularity conditions, we utilize this Kronecker representation to determine
probabilistically the minimal critical points in complexity singly exponential
in the degree of the denominator, with good control over the exponent in the
bit complexity estimate. Generically in the combinatorial case, this allows one
to automatically and rigorously determine asymptotics for the diagonal
coefficient sequence. Examples obtained with a preliminary implementation show
the wide applicability of this approach.Comment: As accepted to proceedings of ISSAC 201
Arithmetic Properties of Picard-Fuchs Equations and Holonomic Recurrences
The coefficient series of the holomorphic Picard-Fuchs differential equation
associated with the periods of elliptic curves often have surprising
number-theoretic properties. These have been widely studied in the case of the
torsion-free, genus zero congruence subgroups of index 6 and 12 (e.g. the
Beauville families). Here, we consider arithmetic properties of the
Picard-Fuchs solutions associated to general elliptic families, with a
particular focus on the index 24 congruence subgroups. We prove that elliptic
families with rational parameters admit linear reparametrizations such that
their associated Picard-Fuchs solutions lie in Z[[t]]. A sufficient condition
is given such that the same holds for holomorphic solutions at infinity. An
Atkin-Swinnerton-Dyer congruence is proven for the coefficient series attached
to \Gamma_1(7). We conclude with a consideration of asymptotics, wherein it is
proved that many coefficient series satisfy asymptotic expressions of the form
u_n \sim \ell \lambda^n/n. Certain arithmetic results extend to the study of
general holonomic recurrences.Comment: 21 pages, to appear in the Journal of Number Theor
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