Results 1  10
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52
The method of creative telescoping
 J. Symbolic Computation
, 1991
"... An algorithm for de6nite hypergeometric summation is given. It is based, in a nonobvious way, on Gosper's algorithm for definite hypergeometric summation, and its theoretical justification relies on Bernstein's theory of holonomic systems. 1. ..."
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Cited by 201 (11 self)
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An algorithm for de6nite hypergeometric summation is given. It is based, in a nonobvious way, on Gosper's algorithm for definite hypergeometric summation, and its theoretical justification relies on Bernstein's theory of holonomic systems. 1.
An Algorithmic Proof Theory for Hypergeometric (ordinary and ``$q$'') Multisum/integral Identities
, 1991
"... this paper we show that these fast algorithms can be extended to the much larger class of multisum terminating hypergeometric (or equivalently, binomial coefficient) identities, to constant term identities of DysonMacdonald type, to MehtaDyson type integrals, and more generally, to identities inv ..."
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Cited by 189 (17 self)
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this paper we show that these fast algorithms can be extended to the much larger class of multisum terminating hypergeometric (or equivalently, binomial coefficient) identities, to constant term identities of DysonMacdonald type, to MehtaDyson type integrals, and more generally, to identities involving any (fixed) number of sums and integrals of products of special functions of hypergeometric type. The computergenerated proofs obtained by our algorithms are always short, are often very elegant, and like the singlesum case, sometimes yield the discovery and proof of new identities. We also do the same for single and multi (terminating) qhypergeometric identities, with continuous and/or discrete variables. Here we describe these algorithms in general, and prove their validity. The validity is an immediate consequence of what we call "The fundamental theorem of hypergeometric summation and integration", a result which we believe is of independent theoretical interest and beauty. The technical aspects of our algorithms, as well as their implementation in Maple, will be described in a forthcoming paper. It is possible, and sometimes preferable, to enjoy a magic show without understanding how the tricks are performed. Hence we invite casual readers to go directly to section 6, in which we give several examples of one or two line proofs generated by our method. In order to understand these proofs, and convince oneself of their correctness, one doesn't need to know how they were generated. Readers can generate many more examples on their own once they obtain a copy of our Maple program, that is available upon request from
Noncommutative Elimination in Ore Algebras Proves Multivariate Identities
 J. SYMBOLIC COMPUT
, 1996
"... ... In this article, we develop a theory of @finite sequences and functions which provides a unified framework to express algorithms proving and discovering multivariate identities. This approach is vindicated by an implementation. ..."
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Cited by 100 (12 self)
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... In this article, we develop a theory of @finite sequences and functions which provides a unified framework to express algorithms proving and discovering multivariate identities. This approach is vindicated by an implementation.
An Extension Of Zeilberger's Fast Algorithm To General Holonomic Functions
 DISCRETE MATH
, 2000
"... We extend Zeilberger's fast algorithm for definite hypergeometric summation to nonhypergeometric holonomic sequences. The algorithm generalizes to differential and qcases as well. Its theoretical justification is based on a description by linear operators and on the theory of holonomy. ..."
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Cited by 90 (5 self)
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We extend Zeilberger's fast algorithm for definite hypergeometric summation to nonhypergeometric holonomic sequences. The algorithm generalizes to differential and qcases as well. Its theoretical justification is based on a description by linear operators and on the theory of holonomy.
D'Alembertian Solutions of Linear Differential and Difference Equations
, 1994
"... D'Alembertian solutions of differential (resp. difference) equations are those expressible as nested indefinite integrals (resp. sums) of hyperexponential functions. They are a subclass of Liouvillian solutions, and can be constructed by recursively finding hyperexponential solutions and reduci ..."
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Cited by 81 (4 self)
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D'Alembertian solutions of differential (resp. difference) equations are those expressible as nested indefinite integrals (resp. sums) of hyperexponential functions. They are a subclass of Liouvillian solutions, and can be constructed by recursively finding hyperexponential solutions and reducing the order. Knowing d'Alembertian solutions of Ly = 0, one can write down the corresponding solutions of Ly = f and of L y = 0. 1 Introduction Functions with rational logarithmic derivative play an important role in manipulation of differential equations, notably in finding Liouvillian solutions of linear differential equations and in factorization of linear differential operators. As any algorithm for finding solutions of linear differential equations in a certain class of functions, the algorithm for finding solutions with rational logarithmic derivative can be combined with the standard technique of reduction of order (also known as d'Alembert substitution) to yield solutions in a (possib...
MultiVariable Zeilberger and AlmkvistZeilberger Algorithms and the Sharpening of WilfZeilberger Theory
 in: Proceedings of FPSAC’09
"... Superficially, this article, dedicated with friendship and admiration to Amitai Regev, has nothing to do with either Polynomial Identity Rings, Representation Theory, or Young tableaux, to all of which he made so many outstanding contributions. But anyone who knows even a little about Amitai Regev’s ..."
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Cited by 44 (10 self)
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Superficially, this article, dedicated with friendship and admiration to Amitai Regev, has nothing to do with either Polynomial Identity Rings, Representation Theory, or Young tableaux, to all of which he made so many outstanding contributions. But anyone who knows even a little about Amitai Regev’s remarkable and versatile research, would know that both sums (and multisums!), and especially integrals (and multiintegrals!) show up very frequently, e.g. see [R], where one of us (DZ) collaborated in the apppendix that consisted in an explicit evaluation of a certain multiintegral. We should also mention that back in the early eighties, Amitai, together with William Beckner [BR], deduced the then wideopen MacdonaldMehta conjecture for the classical root systems BD from Selberg’s integral, a fact that was acknowledged in [M] (albeit with characteristic Macdonaldian understatement). Hence, it is clear that sums, multisums, integrals, and multiintegrals, are Amitai’s bread and butter, and also cup of tea, so the present work has the potential to help him in his future research. A MultiVariable Zeilberger Algorithm Notation. For k integer, (z)k: = z(z + 1)... (z + k − 1), if k ≥ 0 and (z)k: = 1/(z + k)−k if k < 0. In order to avoid too many subscripts in this article, we will denote (z)k by RF (z, k). For
Theorem for a price. Tomorrow’s semirigorous mathematical culture
 MATHEMATICAL INTELLIGENCER
, 1994
"... The most fundamental precept of the mathematical faith is thou shalt prove everything rigorously. While the practitioners of mathematics differ in their view of what constitutes a rigorous proof, and there are fundamentalists who insist on even a more rigorous rigor than the one practiced by the mai ..."
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Cited by 26 (7 self)
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The most fundamental precept of the mathematical faith is thou shalt prove everything rigorously. While the practitioners of mathematics differ in their view of what constitutes a rigorous proof, and there are fundamentalists who insist on even a more rigorous rigor than the one practiced by the mainstream, the belief in this principle could be taken as the defining property of mathematician. The Day After Tomorrow There are writings on the wall that, now that the silicon savior has arrived, a new testament is going to be written. Although there will always be a small group of “rigorous ” oldstyle mathematicians(e.g. [JQ]) who will insist that the true religion is theirs, and that the computer is a false Messiah, they may be viewed by future mainstream mathematicians as a fringe sect of harmless eccentrics, like mathematical physicists are viewed by regular physicists today. The computer has already started doing to mathematics what the telescope and microscope did to astronomy and biology. In the future, not all mathematicians will care about absolute certainty, since there will be so many exciting new facts to discover: mathematical pulsars and quasars that will make the Mandelbrot set seem like a mere Jovian moon. We will have (both human and machine2) professional theoretical mathematicians, who will develop conceptual paradigms to make
Differential equations for algebraic functions
 ISSAC’07: Proceedings of the 2007 international symposium on Symbolic and algebraic computation
, 2007
"... Abstract. It is classical that univariate algebraic functions satisfy linear differential equations with polynomial coefficients. Linear recurrences follow for the coefficients of their power series expansions. We show that the linear differential equation of minimal order has coefficients whose deg ..."
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Cited by 16 (6 self)
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Abstract. It is classical that univariate algebraic functions satisfy linear differential equations with polynomial coefficients. Linear recurrences follow for the coefficients of their power series expansions. We show that the linear differential equation of minimal order has coefficients whose degree is cubic in the degree of the function. We also show that there exists a linear differential equation of order linear in the degree whose coefficients are only of quadratic degree. Furthermore, we prove the existence of recurrences of order and degree close to optimal. We study the complexity of computing these differential equations and recurrences. We deduce a fast algorithm for the expansion of algebraic series. 1.
Multibasic and Mixed Hypergeometric GosperType Algorithms
"... Introduction and notation Let F be a field of characteristic zero and ht n i 1 n=0 a sequence of elements from F which is eventually nonzero. Call t n : 1. hypergeometric, if there are polynomials p 1 ; p 2 2 F [x] such that p 1 (n)t n+1 = p 2 (n)t n for all n; 2. qhypergeometric or basic hyp ..."
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Cited by 15 (0 self)
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Introduction and notation Let F be a field of characteristic zero and ht n i 1 n=0 a sequence of elements from F which is eventually nonzero. Call t n : 1. hypergeometric, if there are polynomials p 1 ; p 2 2 F [x] such that p 1 (n)t n+1 = p 2 (n)t n for all n; 2. qhypergeometric or basic hypergeometric, if there are polynomials p 1 ; p 2 2 F [x] such that p 1 (q n )t n+1 = p 2 (q n<