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THE WZ METHOD AND FLAWLESS WZ PAIRS

Part of: Computational aspects Sequences and sets

Published online by Cambridge University Press:  28 April 2025

JESÚS GUILLERA Open the ORCID record for JESÚS GUILLERA [Opens in a new window]
JESÚS GUILLERA*
Affiliation:
Department of Mathematics, University of Zaragoza, 50009 Zaragoza, Spain
*
e-mail: [email protected]
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Abstract

Kam Cheong Au [‘Wilf–Zeilberger seeds and non-trivial hypergeometric series’, Journal of Symbolic Computation 130 (2025), Article no. 102241] discovered a powerful methodology for finding new Wilf–Zeilberger (WZ) pairs. He calls it WZ seeds and gives numerous examples of applications to proving longstanding conjectural identities for reciprocal powers of $\pi $ and their duals for Dirichlet L-values. In this note, we explain how a modification of Au’s WZ pairs together with a classical analytic argument yields simpler proofs of these results. We illustrate our method with examples elaborated with assistance of Maple code that we have developed.

Keywords

WZ pairRamanujan-like series

MSC classification

Primary: 33F10: Symbolic computation (Gosper and Zeilberger algorithms, etc.)
Secondary: 11B65: Binomial coefficients; factorials; $q$-identities 33C20: Generalized hypergeometric series, $_pF_q$
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 9
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc

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Footnotes

Dedicated to Doron Zeilberger on his 75th birthday

References

Almkvist, G., ‘Glaisher’s formulas for $1/{\pi}^2$ and some generalizations’, in: Advances in Combinatorics (eds. Kotsireas, I. S. and Zima, E. V.) (Springer, Heidelberg, 2013), 1–21; with an Appendix by Arne Meurman.Google Scholar
Au, K. C., ‘Wilf–Zeilberger seeds: $q$ -analogues’, Preprint, 2024, arXiv:2403.04555; APPENDIX B.Google Scholar
Au, K. C., ‘Wilf–Zeilberger seeds and non-trivial hypergeometric series’, J. Symbolic Comput. 130 (2025), Article no. 102241.CrossRefGoogle Scholar
Ekhad, S. and Zeilberger, D., ‘A WZ proof of Ramanujan’s formula for Pi’, in: Geometry, Analysis and Mechanics (ed. J. M. Rassias) (World Scientific, Singapore, 1994), 107–108.CrossRefGoogle Scholar
Guillera, J., ‘About a new kind of Ramanujan-type series’, Exp. Math. 12 (2003), 507–510.CrossRefGoogle Scholar
Guillera, J., ‘Generators of some Ramanujan formulas’, Ramanujan J. 111 (2006), 41–48.CrossRefGoogle Scholar
Guillera, J., ‘On WZ pairs which prove Ramanujan series’, Ramanujan J. 22 (2010), 249–259.CrossRefGoogle Scholar
Guillera, J., ‘WZ-proofs of “divergent” Ramanujan-type series’, in: Advances in Combinatorics (eds. Kotsireas, I. S. and Zima, E. V.) (Springer, Heidelberg, 2013), 187–195.CrossRefGoogle Scholar
Guillera, J., Maple program: On WZ pairs which prove Ramanujan series (2024), updates in progress; https://anamat.unizar.es/jguillera/Maple.html.Google Scholar
Guillera, J., ‘Heuristic derivation of Zudilin’s supercongruences for rational Ramanujan series’, Rev. Mat. Complut. 38 (2025), 295–305.CrossRefGoogle Scholar
Petkovšek, M., Wilf, H. and Zeilberger, D., A=B (A. K. Peters, New York, 1996).CrossRefGoogle Scholar
Wilf, H. and Zeilberger, D., ‘Rational functions certify combinatorial identities’, J. Amer. Math. Soc. 3 (1990), 147–158 (winner of the Steel Prize).CrossRefGoogle Scholar