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How to Generate All Possible Rational Wilf-Zeilberger Pairs?

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Algorithms and Complexity in Mathematics, Epistemology, and Science

Part of the book series: Fields Institute Communications ((FIC,volume 82))

Abstract

A Wilf–Zeilberger pair (F, G) in the discrete case satisfies the equation

$$\displaystyle F(n+1, k) - F(n, k) = G(n, k+1) - G(n, k). $$

We present a structural description of all possible rational Wilf–Zeilberger pairs and their continuous and mixed analogues.

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Notes

  1. 1.

    The talk was given at the Waterloo Workshop in Computer Algebra (in honor of Herbert Wilf’s 80th birthday), Wilfrid Laurier University, May 28, 2011. For the talk slides, see the link: http://people.brandeis.edu/~gessel/homepage/slides/wilf80-slides.pdf.

  2. 2.

    This is motivated by the fact that a differential form ω = gdx + fdy with f, g ∈ K(x, y) is exact in K(x, y) if and only if f = D y(h) and g = D x(h) for some h ∈ K(x, y).

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Acknowledgements

I would like to thank Prof. Victor J.W. Guo and Prof. Zhi-Wei Sun for many discussions on series for special constants, (super)-congruences and their q-analogues that can be proved using the WZ method. I am also very grateful to Ruyong Feng and Rong-Hua Wang for many constructive comments on the earlier version of this paper. I also thank the anonymous reviewers for their constructive and detailed comments.This work was supported by the NSFC grants 11501552, 11688101 and by the Frontier Key Project (QYZDJ-SSW-SYS022) and the Fund of the Youth Innovation Promotion Association, CAS.

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Authors and Affiliations

  1. KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China

    Shaoshi Chen

  2. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China

    Shaoshi Chen

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  1. Shaoshi Chen
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Correspondence to Shaoshi Chen .

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Editors and Affiliations

  1. Department of Philosophy, Simon Fraser University Philosophy, Burnaby, BC, Canada

    Nicolas Fillion

  2. The Rotman Institute of Philosophy, The Ontario Research Center for Computer Algebra and The School of Mathematical and Statistical Sciences, The University of Western Ontario, London, ON, Canada

    Robert M. Corless

  3. Department of Physics & Computer Science, Wilfrid Laurier University, Waterloo, ON, Canada

    Ilias S. Kotsireas

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Dedicated to the memory of Jonathan M. Borwein and Ann Johnson.

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Chen, S. (2019). How to Generate All Possible Rational Wilf-Zeilberger Pairs?. In: Fillion, N., Corless, R., Kotsireas, I. (eds) Algorithms and Complexity in Mathematics, Epistemology, and Science. Fields Institute Communications, vol 82. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-9051-1_2

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