Results 1  10
of
65
Liouvillian solutions of linear differential equations with liouvillian coefficients
 Computers and Mathematics
, 1989
"... Let L(y) = b be a linear differential equation with coefficients in a differential field K. We discuss the problem of deciding if such an equation has a nonzero solution in K and give a decision procedure in case K is an elementary extension of the field of rational functions or is an algebraic ex ..."
Abstract

Cited by 52 (5 self)
 Add to MetaCart
Let L(y) = b be a linear differential equation with coefficients in a differential field K. We discuss the problem of deciding if such an equation has a nonzero solution in K and give a decision procedure in case K is an elementary extension of the field of rational functions or is an algebraic extension of a transcendental liouvillian extension of the field of rational functions. We show how one can use this result to give a procedure to find a basis for the space of solutions, liouvillian over K, of L(y)=0 where K is such a field and L(y) has coefficients in K. 1.
The method of differentiating under the integral sign
 J. Symbolic Computation
, 1990
"... ”I could never resist a definite integral ” (G. H. Hardy [Co]) ”One thing I never did learn was contour integration. I had learned to do integrals by various methods shown in a book that my highschool physics teacher Mr. Bader had given me.... The book also showed how to differentiate parameters un ..."
Abstract

Cited by 46 (16 self)
 Add to MetaCart
(Show Context)
”I could never resist a definite integral ” (G. H. Hardy [Co]) ”One thing I never did learn was contour integration. I had learned to do integrals by various methods shown in a book that my highschool physics teacher Mr. Bader had given me.... The book also showed how to differentiate parameters under the integral signIt’s a certain operation. It turns out that’s not taught very much in the universities; they don’t emphasize it. But I caught on how to use that method, and I used that one damn tool again and again. So because I was selftaught using that book, I had peculiar methods of doing integrals. The result was that, when guys at MIT or Princeton had trouble doing a certain integral, it was because they couldn’t do it with the standard methods they had learned in school. If it was contour integration, they would have found it; if it was a simple series expansion, they would have found it. Then I come along and try differentiating under the integral sign, and often it worked. So I got a great reputation for doing integrals, only because my box of tools was different from everybody else’s, and they had tried all their tools on it before giving the problem to me. ” (Richard P. Feynman[F])
The summation package Sigma: Underlying principles and a rhombus tiling application
 Discrete Math. Theor. Comput. Sci
, 2004
"... We give an overview of how a huge class of multisum identities can be proven and discovered with the summation package Sigma implemented in the computer algebra system Mathematica. General principles of symbolic summation are discussed. We illustrate the usage of Sigma by showing how one can find an ..."
Abstract

Cited by 37 (28 self)
 Add to MetaCart
We give an overview of how a huge class of multisum identities can be proven and discovered with the summation package Sigma implemented in the computer algebra system Mathematica. General principles of symbolic summation are discussed. We illustrate the usage of Sigma by showing how one can find and prove a multisum identity that arose in the enumeration of rhombus tilings of a symmetric hexagon. Whereas this identity has been derived alternatively with the help of highly involved transformations of special functions, our tools enable to find and prove this identity completely automatically with the computer.
Solving parameterized linear difference equations in terms of indefinite nested sums and products
 J. Differ. Equations Appl
, 2002
"... The described algorithms enable one to find all solutions of parameterized linear difference equations within ΠΣfields, a very general class of difference fields. These algorithms can be applied to a very general class of multisums, for instance, for proving identities and simplifying expressions. ..."
Abstract

Cited by 35 (25 self)
 Add to MetaCart
(Show Context)
The described algorithms enable one to find all solutions of parameterized linear difference equations within ΠΣfields, a very general class of difference fields. These algorithms can be applied to a very general class of multisums, for instance, for proving identities and simplifying expressions. Keywords: symbolic summation, difference fields, ΠΣextensions, ΠΣfields AMS Subject Classification: 33FXX, 68W30, 12H10 1
Padé approximations to the logarithm III: Alternative methods and additional results
 Ramanujan J
, 2004
"... Abstract. We use C. Schneider’s summation software Sigma and a method due to AndrewsNewtonZeilberger to reprove results from [5] and [24] as well as to prove new related material. 1. ..."
Abstract

Cited by 21 (13 self)
 Add to MetaCart
(Show Context)
Abstract. We use C. Schneider’s summation software Sigma and a method due to AndrewsNewtonZeilberger to reprove results from [5] and [24] as well as to prove new related material. 1.
Rational General Solutions of Algebraic Ordinary Differential Equations
 Proc. ISSAC2004
, 2004
"... We give a necessary and sufficient condition for an algebraic ODE to have a rational type general solution. For an autonomous first order ODE, we give an algorithm to compute a rational general solution if it exists. The algorithm is based on the relation between rational solutions of the first or ..."
Abstract

Cited by 16 (2 self)
 Add to MetaCart
(Show Context)
We give a necessary and sufficient condition for an algebraic ODE to have a rational type general solution. For an autonomous first order ODE, we give an algorithm to compute a rational general solution if it exists. The algorithm is based on the relation between rational solutions of the first order ODE and rational parametrizations of the plane algebraic curve defined by the first order ODE and Padé approximants.
SYMBOLIC SUMMATION ASSISTS COMBINATORICS
 SÉMINAIRE LOTHARINGIEN DE COMBINATOIRE 56 (2007), ARTICLE B56B
, 2007
"... We present symbolic summation tools in the context of difference fields that help scientists in practical problem solving. Throughout this article we present multisum examples which are related to combinatorial problems. ..."
Abstract

Cited by 15 (8 self)
 Add to MetaCart
We present symbolic summation tools in the context of difference fields that help scientists in practical problem solving. Throughout this article we present multisum examples which are related to combinatorial problems.
A refined difference field theory for symbolic summation
 Journal of Symbolic Computation
, 2008
"... In this article we present a refined summation theory based on Karr’s difference field approach. The resulting algorithms find sum representations with optimal nested depth. For instance, the algorithms have been applied successively to evaluate Feynman integrals from Perturbative Quantum Field Theo ..."
Abstract

Cited by 13 (8 self)
 Add to MetaCart
(Show Context)
In this article we present a refined summation theory based on Karr’s difference field approach. The resulting algorithms find sum representations with optimal nested depth. For instance, the algorithms have been applied successively to evaluate Feynman integrals from Perturbative Quantum Field Theory. Key words: Symbolic summation, difference fields, nested depth
Computing monodromy groups defined by plane algebraic curves
 In: Proceedings of the 2007 International Workshop on Symbolicnumeric Computation. ACM, NewYork
, 2007
"... We present a symbolicnumeric method to compute the monodromy group of a plane algebraic curve viewed as a ramified covering space of the complex plane. Following the definition, our algorithm is based on analytic continuation of algebraic functions above paths in the complex plane. Our contribution ..."
Abstract

Cited by 12 (4 self)
 Add to MetaCart
(Show Context)
We present a symbolicnumeric method to compute the monodromy group of a plane algebraic curve viewed as a ramified covering space of the complex plane. Following the definition, our algorithm is based on analytic continuation of algebraic functions above paths in the complex plane. Our contribution is threefold: first of all, we show how to use a minimum spanning tree to minimize the length of paths; then, we propose a strategy that gives a good compromise between the number of steps and the truncation orders of Puiseux expansions, obtaining for the first time a complexity result about the number of steps; finally, we present an efficient numericalmodular algorithm to compute Puiseux expansions above critical points, which is a non trivial task.