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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 ..."
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Cited by 59 (18 self)
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”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])
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 ..."
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Cited by 57 (6 self)
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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.
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. ..."
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Cited by 44 (29 self)
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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
Computer Proofs of a New Family of Harmonic Number Identities
, 2002
"... In this paper we consider five conjectured harmonic number identities similar to those arising in the context of supercongruences for Apéry numbers. The general object of this article is to discuss the possibility of automating not only the proof but also the discovery of such formulas. As a specifi ..."
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Cited by 42 (23 self)
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In this paper we consider five conjectured harmonic number identities similar to those arising in the context of supercongruences for Apéry numbers. The general object of this article is to discuss the possibility of automating not only the proof but also the discovery of such formulas. As a specific application we consider two different algorithmic methods to derive and to prove the five conjectured identities. One is based on an extension of Karr’s summation algorithm in difference fields. The other method combines an old idea of Newton (which has been extended by Andrews) with Zeilberger’s algorithm for definite hypergeometric sums.
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 ..."
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Cited by 39 (29 self)
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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.
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. ..."
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Cited by 26 (15 self)
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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.
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 ..."
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Cited by 24 (12 self)
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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
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. ..."
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Cited by 24 (11 self)
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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.