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Telescoping in the context of symbolic summation in Maple

S.A. Abramov, J.J. Carette, K.O. Geddes, H.Q. Le
2004 Journal of symbolic computation  
A design and an implementation of a software package which incorporates these methods into the computer algebra system Maple are described in detail.  ...  The focus is on recently-developed results on computing closed forms of definite sums of hypergeometric terms.  ...  The end result is the software package SumTools, a symbolic summation toolbox in Maple. Symbolic summation is a vast research area in computer algebra.  ... 
doi:10.1016/j.jsc.2003.08.010 fatcat:lnqitggi7rf5xnash4yaixmwau

Index to Volumes 37 and 38

2004 Journal of symbolic computation  
, R.S. and MILLER, J.C., Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDEs, 669 BALLICO, E. and ORECCHIA, F., Computing minimal generators of the  ...  test for (i, j )-uniformity in reduced zero-dimensional schemes, 403 AHN, M.  ...  coefficient bounds, 1507 Special polycyclic generating sequences for finite soluble groups, 1445 Telescoping in the context of symbolic summation in Maple, 1303 The Cartan matrix and enumerative calculus  ... 
doi:10.1016/s0747-7171(04)00109-9 fatcat:q3cckydpknhjhinygacsvlj52y

Creative Telescoping for Holonomic Functions [chapter]

Christoph Koutschan
2013 Texts & Monographs in Symbolic Computation  
the impact of creative telescoping in numerous contexts.  ...  On the other hand, this chapter has the flavour of a survey article: the developments in this area during the last two decades are sketched and a selection of references is compiled in order to highlight  ...  In the setting of difference fields, Schneider developed a sophisticated symbolic summation theory [86] whose core again is creative telescoping.  ... 
doi:10.1007/978-3-7091-1616-6_7 fatcat:7kmlabkcejfsvpwcjfi6vwrth4

A Computer-Algebra-Based Formal Proof of the Irrationality of ζ(3) [chapter]

Frédéric Chyzak, Assia Mahboubi, Thomas Sibut-Pinote, Enrico Tassi
2014 Lecture Notes in Computer Science  
We formally prove this result by an a posteriori verification of calculations performed by computer algebra algorithms in a Maple session.  ...  This result was first proved by Apéry in 1978, and the proof we have formalized follows the path of his original presentation.  ...  The most notable example being search, that is almost ineffective in the context of the algebraic hierarchy.  ... 
doi:10.1007/978-3-319-08970-6_11 fatcat:mlorqqxcxbb7dpjx5i53dfmudu

Greatest Factorial Factorization and Symbolic Summation

Peter Paule
1995 Journal of symbolic computation  
We illustrate the fundamental role of that concept in the context of symbolic summation.  ...  Besides a detailed discussion of the basic GFF notions we present a new approach to the inde nite rational summation problem as well as to Gosper's algorithm for summing hypergeometric sequences.  ...  One of his valuable suggestions for streamlining the presentation was to introduce the reduction Lemma 3.2.  ... 
doi:10.1006/jsco.1995.1049 fatcat:qha3kq5w6nbkflxuy6qdvwifyu

Creative Telescoping on Multiple Sums [article]

Christoph Koutschan, Elaine Wong
2021 arXiv   pre-print
In particular, we highlight the necessity of the certificate in these computations and how its complexity can greatly influence the computation time.  ...  In other words, no creative telescoping implementation currently exists that can resolve all these issues automatically.  ...  We would especially like to acknowledge Pierre Lairez for pointing us to his paper [2] and for demonstrating how to make the computation in Section 3.8 with his binomial sums Maple package.  ... 
arXiv:2010.08889v2 fatcat:wk5lkno4obda3pw7adjknzb56q

-XSummer- Transcendental functions and symbolic summation in Form

S. Moch, P. Uwer
2006 Computer Physics Communications  
Harmonic sums and their generalizations are extremely useful in the evaluation of higher-order perturbative corrections in quantum field theory.  ...  In this Letter we discuss the implementation of several algorithms to solve these sums by algebraic means, using the computer algebra system Form.  ...  The work of S.M. has been supported in part by the Helmholtz Gemeinschaft under contract VH-NG-105 and by the Deutsche Forschungsgemeinschaft in Sonderforschungsbereich/Transregio 9.  ... 
doi:10.1016/j.cpc.2005.12.014 fatcat:qswxn37lanazfp7crtaoky7h3u

Creative Telescoping for Parametrised Integration and Summation

Frédéric Chyzak
2011 Les cours du CIRM  
II-9 Course n o II-Creative Telescoping for Parametrised Integration and Summation CHAPTER 2 Early History of Creative Telescoping Before the technical results in the next chapters, I provide a brief  ...  II-13 Course n o II-Creative Telescoping for Parametrised Integration and Summation CHAPTER 3 Creative-Telescoping Algorithms for Equations of the First Order 1.  ... 
doi:10.5802/ccirm.14 fatcat:5uu7v6zlljda7jlbbqmfxostte

A series evaluation technique based on a modified Abel lemma

JOHN MAXWELL CAMPBELL, MARCO CANTARINI
2022 Turkish Journal of Mathematics  
We introduce a technique for determining infinite series identities through something of a combination of the modified Abel lemma on summation by parts and a method of undetermined coefficients.  ...  We succeed in applying our technique in our proving a nontrivial variant of Gauss' hypergeometric identity, giving us an evaluation for a family of 3F2(1) -series with three free parameters, and to establish  ...  The second author is the member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).  ... 
doi:10.55730/1300-0098.3177 fatcat:xmfb6uv4fveqjkxnvwqjyzqgj4

Definite Sums of Hypergeometric Terms and Limits of P-Recursive Sequences [article]

Hui Huang
2017 arXiv   pre-print
The ubiquity of the class of D-finite functions and P-recursive sequences in symbolic computation is widely recognized. In this thesis, the presented work consists of two parts related to this class.  ...  In the first part, we generalize the reduction-based creative telescoping algorithms to the hypergeometric setting, which allows to deal with definite sums of hypergeometric terms more quickly.  ...  In the study of symbolic summation, there are mainly two kinds of problems related to hypergeometric terms. Problem 1.1 (Hypergeometric summation).  ... 
arXiv:1710.08566v1 fatcat:he6mnewnd5cfdojbymo5byxpfa

Ten Problems in Experimental Mathematics

David H. Bailey, Jonathan M. Borwein, Vishaal Kapoor, Eric W. Weisstein
2006 The American mathematical monthly  
It should be mentioned that both Maple and the ISC identified the constant ζ(3, 1) (see the remark under the "history and context" heading).  ...  In such cases, the Euler-Maclaurin summation formula [2, p. 180] can be invoked to conclude that the error E(h) in the approximation (28) decreases very rapidly-faster than any power of h.  ...  which reduces the given double Euler sums to a sum of products of classical ζ-values. Euler also noted the first reflection formulas certainly valid when a > 1 and b > 1.  ... 
doi:10.1080/00029890.2006.11920330 fatcat:7biryus4jvckjbraro3d3h2afm

Ten Problems in Experimental Mathematics

David H. Bailey, Jonathan M. Borwein, Vishaal Kapoor, Eric W. Weisstein
2006 The American mathematical monthly  
It should be mentioned that both Maple and the ISC identified the constant ζ(3, 1) (see the remark under the "history and context" heading).  ...  In such cases, the Euler-Maclaurin summation formula [2, p. 180] can be invoked to conclude that the error E(h) in the approximation (28) decreases very rapidly-faster than any power of h.  ...  which reduces the given double Euler sums to a sum of products of classical ζ-values. Euler also noted the first reflection formulas certainly valid when a > 1 and b > 1.  ... 
doi:10.2307/27641975 fatcat:p5deks7g7jepfh4u2g2tmzyh64

Deconstructing the Zeilberger algorithm†

Doron Zeilberger
2005 Journal of difference equations and applications (Print)  
Hence WZ theory is transgressing the boundaries of the context of justification into the context of discovery.  ...  The accompanying Maple package FindHypergeometric explains the various miracles that account for the classical evaluations, and the more specialized Maple package twoFone, also accompanying this article  ...  In particular, the Zeilberger algorithm (as implemented in my own Maple package EKHAD, and starting with Maple 6, in the built-in package SumTools[Hypergeometric], and there is a very popular Mathematica  ... 
doi:10.1080/10236190500089911 fatcat:ckqbmjeexbdg3ereoge7xy65de

Automatic Improvements of Wallis' Inequality

P Paule, V Pillwein
2010 2010 12th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing  
In the main part of the article we automatically refine the bounds in Wallis' inequality by using methods based on Collins' cylindrical algebraic decomposition.  ...  The mathematical backbone of this article is formed by three classical formulas of Wallis: his product formula for π, an inequality implying the product formula in the limit, and a related definite integral  ...  Evaluation by creative telescoping The definite integration problem can be transformed into a definite summation problem.  ... 
doi:10.1109/synasc.2010.89 dblp:conf/synasc/PauleP10 fatcat:bs2yoc3khra37bjqv4t64wm5eu

Efficient Algorithms for Mixed Creative Telscoping

Alin Bostan, Louis Dumont, Bruno Salvy
2016 Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation - ISSAC '16  
Its analysis reveals tight bounds on the sizes of the telescoper it produces.  ...  Creative telescoping is a powerful computer algebra paradigm -initiated by Doron Zeilberger in the 90's- for dealing with definite integrals and sums with parameters.  ...  We are grateful to the referees for their thorough work and helpful comments. This work has been supported in part by FastRelax ANR-14-CE25-0018-01.  ... 
doi:10.1145/2930889.2930907 dblp:conf/issac/BostanDS16 fatcat:kzmyji4yz5f5pfnvqtxfhllv7q
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