research-article

Representation of hypergeometric products in difference rings

Authors:

Evans Doe Ocansey,

Carsten SchneiderAuthors Info & Claims

ACM Communications in Computer Algebra, Volume 50, Issue 4

Pages 161 - 163

https://doi.org/10.1145/3055282.3055290

Published: 22 February 2017 Publication History

Abstract

In his pioneering work [1, 2], Michael Karr introduced ΠΣ-fields which provide a rather general framework for symbolic summation. He worked out the first algorithmic steps to represent indefinite nested sums and products as transcendental extensions over a computable ground field K called the field of constants. Furthermore, he presented an algorithm that solves the parameterized telescoping problem, and as special cases the telescoping and creative telescoping problems [3] within a given ΠΣ-field.

References

[1]

Michael Karr. Summation in Finite Terms. Journal of the ACM (JACM),28(2):305--350, 1981.

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[2]

Michael Karr. Theory of Summation in Finite Terms, J. Symbolic Comput.,1:303--315, 1985.

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[3]

Doron Zeilberger. The Method of Creative Telescoping. J. Symbolic Comput., 11:195--204, 1991.

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[4]

Carsten Schneider. A Streamlined Difference Ring Theory: Indefinite Nested Sums, the Alternating Sign, and the Parameterized Telescoping Problem, 2014 16th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC), 26--33, 2014.

[5]

Carsten Schneider. A Difference Ring Theory for Symbolic Summation, J. Symbolic Comput.,72, 82--127, 2016.

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[6]

Carsten Schneider. Product Representations in ΠΣ-Fields, Annals of Combinatorics, 9(1):75--99, 2005.

[7]

Marius van der Put and Michael F. Singer. Galois Theory of Difference Equations, volume 1666 of Lecture Notes in Mathematics., Springer-Verlag, Berlin,1997.

[8]

Carsten Schneider. Summation Theory II: Characterizations of RΠΣ*-Extensions and Algorithmic Aspects. J. Symbolic Comput., in press, 2016.

[9]

Carsten Schneider. Symbolic Summation in Difference Rings and Applications. In M. Rosenkranz, editor Proc. ISSAC 2016, 2016.

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[10]

Sergei A. Abramov and Marko Petkovšek. Polynomial Ring Automorphisms, Rational (w, σ)- Canonical Forms, and the Assignment Problem, J. Symbolic Comput., 45(6):684--708, 2010.

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[11]

Guoqiang Ge. Algorithms Related to the Multiplicative Representation of Algebraic Numbers, PhD thesis, Univeristy of California at Berkeley, 1993.

[12]

Manuel Kauers. Algorithms for Nonlinear Higher Order Difference Equations. PhD thesis, RISC-Linz, October 2005.

[13]

Andrej Bauer and Marko Petkovšek. Multibasic and Mixed Hypergeometric Gosper-Type Algorithms. J. Symbolic Comput., 28:(4--5):711--736, October, 1999.

[14]

Carsten Schneider. Symbolic Summation Assists Combinatorics. Sém. Lothar. Combin., 56:1--36, 2007. Article B56b.

[15]

Shaoshi Chen and Frédéric Chyzak and Ruyong Feng and Guofeng Fu and Ziming Li. On the Existence of Telescopers for Mixed Hypergeometric Terms. J. Symbolic Comput., 68:1--26, 2015.

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Cited By

Schneider CAbramov SZima EGao X(2016)Symbolic Summation in Difference Rings and ApplicationsProceedings of the ACM on International Symposium on Symbolic and Algebraic Computation10.1145/2930889.2930945(9-12)Online publication date: 20-Jul-2016
https://dl.acm.org/doi/10.1145/2930889.2930945

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Published In

cover image ACM Communications in Computer Algebra

ACM Communications in Computer Algebra Volume 50, Issue 4

December 2016

66 pages

ISSN:1932-2232

EISSN:1932-2240

DOI:10.1145/3055282

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Copyright © 2017 Authors.

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 22 February 2017

Published in SIGSAM-CCA Volume 50, Issue 4

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Cited By

Schneider CAbramov SZima EGao X(2016)Symbolic Summation in Difference Rings and ApplicationsProceedings of the ACM on International Symposium on Symbolic and Algebraic Computation10.1145/2930889.2930945(9-12)Online publication date: 20-Jul-2016
https://dl.acm.org/doi/10.1145/2930889.2930945

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