72 research outputs found
Bivariate Extensions of Abramov's Algorithm for Rational Summation
Abramov's algorithm enables us to decide whether a univariate rational
function can be written as a difference of another rational function, which has
been a fundamental algorithm for rational summation. In 2014, Chen and Singer
generalized Abramov's algorithm to the case of rational functions in two
(-)discrete variables. In this paper we solve the remaining three mixed
cases, which completes our recent project on bivariate extensions of Abramov's
algorithm for rational summation.Comment: Dedicated to Professor Sergei A. Abramov on the occasion of his 70th
birthda
A Note on Lipshitz's Lemma 3
In this note, we give a remark on the proof of Lemma 3 in Lipshitz's paper
"The diagonal of a D-Finite power series is D-Finite". This remark is motivated
by the observation that the statement from line -8 to -3 on page 375 of that
paper seems not completely correct.Comment: 5 page
Order-Degree Curves for Hypergeometric Creative Telescoping
Creative telescoping applied to a bivariate proper hypergeometric term
produces linear recurrence operators with polynomial coefficients, called
telescopers. We provide bounds for the degrees of the polynomials appearing in
these operators. Our bounds are expressed as curves in the (r,d)-plane which
assign to every order r a bound on the degree d of the telescopers. These
curves are hyperbolas, which reflect the phenomenon that higher order
telescopers tend to have lower degree, and vice versa
Residues and Telescopers for Rational Functions
We give necessary and sufficient conditions for the existence of telescopers
for rational functions of two variables in the continuous, discrete and
q-discrete settings and characterize which operators can occur as telescopers.
Using this latter characterization, we reprove results of Furstenberg and
Zeilberger concerning diagonals of power series representing rational
functions. The key concept behind these considerations is a generalization of
the notion of residue in the continuous case to an analogous concept in the
discrete and q-discrete cases.Comment: 30 page
Power Series with Coefficients from a Finite Set
We prove in this paper that a multivariate D-finite power series with
coefficients from a finite set is rational. This generalizes a rationality
theorem of van der Poorten and Shparlinski in 1996.Comment: 11 page
Desingularization of Ore Operators
We show that Ore operators can be desingularized by calculating a least
common left multiple with a random operator of appropriate order. Our result
generalizes a classical result about apparent singularities of linear
differential equations, and it gives rise to a surprisingly simple
desingularization algorithm
Additive Decompositions in Primitive Extensions
This paper extends the classical Ostrogradsky-Hermite reduction for rational
functions to more general functions in primitive extensions of certain types.
For an element in such an extension , the extended reduction decomposes
as the sum of a derivative in and another element such that has
an antiderivative in if and only if ; and has an elementary
antiderivative over if and only if is a linear combination of
logarithmic derivatives over the constants when is a logarithmic extension.
Moreover, is minimal in some sense. Additive decompositions may lead to
reduction-based creative-telescoping methods for nested logarithmic functions,
which are not necessarily -finite
On the Structure of Compatible Rational Functions
A finite number of rational functions are compatible if they satisfy the
compatibility conditions of a first-order linear functional system involving
differential, shift and q-shift operators. We present a theorem that describes
the structure of compatible rational functions. The theorem enables us to
decompose a solution of such a system as a product of a rational function,
several symbolic powers, a hyperexponential function, a hypergeometric term,
and a q-hypergeometric term. We outline an algorithm for computing this
product, and present an application
Apparent Singularities of D-finite Systems
We generalize the notions of singularities and ordinary points from linear
ordinary differential equations to D-finite systems. Ordinary points of a
D-finite system are characterized in terms of its formal power series
solutions. We also show that apparent singularities can be removed like in the
univariate case by adding suitable additional solutions to the system at hand.
Several algorithms are presented for removing and detecting apparent
singularities. In addition, an algorithm is given for computing formal power
series solutions of a D-finite system at apparent singularities
Complexity of Creative Telescoping for Bivariate Rational Functions
The long-term goal initiated in this work is to obtain fast algorithms and
implementations for definite integration in Almkvist and Zeilberger's framework
of (differential) creative telescoping. Our complexity-driven approach is to
obtain tight degree bounds on the various expressions involved in the method.
To make the problem more tractable, we restrict to bivariate rational
functions. By considering this constrained class of inputs, we are able to
blend the general method of creative telescoping with the well-known Hermite
reduction. We then use our new method to compute diagonals of rational power
series arising from combinatorics.Comment: 8 page
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