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``Divergent'' Ramanujan-type supercongruences


Authors: Jesús Guillera and Wadim Zudilin
Journal: Proc. Amer. Math. Soc. 140 (2012), 765-777
MSC (2010): Primary 11Y55, 33C20, 33F10; Secondary 11B65, 11D88, 11F33, 11F85, 11S80, 12H25, 40G99, 65-05, 65B10
DOI: https://doi.org/10.1090/S0002-9939-2011-10950-X
Published electronically: July 21, 2011
MathSciNet review: 2869062
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Abstract | References | Similar Articles | Additional Information

Abstract: ``Divergent'' Ramanujan-type series for $ 1/\pi$ and $ 1/\pi^2$ provide us with new nice examples of supercongruences of the same kind as those related to the convergent cases. In this paper we manage to prove three of the supercongruences by means of the Wilf-Zeilberger algorithmic technique.


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  • 1. G. ALMKVIST and A. GRANVILLE, Borwein and Bradley's Apéry-like formulae for $ \zeta(4n+3)$, Experiment. Math. 8 (1999), 197-203. MR 1700578 (2000h:11126)
  • 2. J. M. BORWEIN and P. B. BORWEIN, More Ramanujan-type series for $ 1/\pi$, in Ramanujan Revisited, G. E. Andrews et al. (eds.) (Academic Press, Boston, 1988), 359-374. MR 938974 (89d:11118)
  • 3. J. M. BORWEIN and D. M. BRADLEY, Empirically determined Apéry-like formulae for $ \zeta(4n+3)$, Experiment. Math. 6 (1997), 181-194. MR 1481588 (98m:11142)
  • 4. H. H. CHAN, S. H. CHAN, and Z. G. LIU, Domb's numbers and Ramanujan-Sato type series for $ 1/\pi$, Adv. in Math. 186 (2004), 396-410. MR 2073912 (2005e:11170)
  • 5. H. H. CHAN and W. ZUDILIN, New representations for Apéry-like sequences, Mathematika 56:1 (2010), 107-117. MR 2604987
  • 6. J. GUILLERA, Hypergeometric identities for 10 extended Ramanujan-type series, Ramanujan J. 15:2 (2008), 219-234. MR 2377577 (2009m:33014)
  • 7. J. GUILLERA, A matrix form of Ramanujan-type series for $ 1/\pi$, in Gems in Experimental Mathematics, T. Amdeberhan, L. A. Medina, and V. H. Moll (eds.), Contemp. Math. 517 (Amer. Math. Soc., Providence, RI, 2010), 189-206. MR 2731088
  • 8. J. GUILLERA, WZ-proofs of ``divergent'' Ramanujan-type series, preprint at arXiv: 1012.2681 (2010).
  • 9. J. GUILLERA, Mosaic supercongruences of Ramanujan type, preprint at arXiv: 1007.2290 (2010). To appear in Experiment. Math.
  • 10. F. MORLEY, Note on the congruence $ 2^{4n}\equiv(-1)^n(2n)!/(n!)^2$, where $ 2n+1$ is a prime, Ann. Math. 9 (1895), 168-170. MR 1502188
  • 11. M. PETKOVŠEK, H. S. WILF, and D. ZEILBERGER, $ A=B$ (A K Peters, Ltd., Wellesley, MA, 1997). MR 1379802 (97j:05001)
  • 12. L. J. SLATER, Generalized hypergeometric functions (Cambridge University Press, Cambridge, 1966). MR 0201688 (34:1570)
  • 13. T. B. STAVER, Om summasjon av potenser av binomialkoeffisienten, Norsk Mat. Tidsskrift 29 (1947), 97-103. MR 0024874 (9:560a)
  • 14. ZHI-WEI SUN, Super congruences and Euler numbers, preprint at arXiv: 1001.4453 (2010).
  • 15. ZHI-WEI SUN and R. TAURASO, New congruences for central binomial coefficients, Adv. in Appl. Math. 45:1 (2010), 125-148. MR 2628791
  • 16. R. TAURASO, More congruences for central binomial coefficients, J. Number Theory 130:12 (2010), 2639-2649. MR 2684486
  • 17. W. ZUDILIN, Quadratic transformations and Guillera's formulae for $ 1/\pi^2$, Mat. Zametki 81:3 (2007), 335-340; English transl., Math. Notes 81:3 (2007), 297-301. MR 2333939 (2008k:33025)
  • 18. W. ZUDILIN, Ramanujan-type formulae for $ 1/\pi$: A second wind?, in Modular Forms and String Duality (Banff, June 3-8, 2006), N. Yui, H. Verrill, and C. F. Doran (eds.), Fields Inst. Commun. Ser. 54 (Amer. Math. Soc. & Fields Inst., 2008), 179-188. MR 2454325 (2010f:11183)
  • 19. W. ZUDILIN, Ramanujan-type supercongruences, J. Number Theory 129:8 (2009), 1848-1857. MR 2522708

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Additional Information

Jesús Guillera
Affiliation: Av. Cesáreo Alierta, 31 esc. izda 4$^{∘}$–A, Zaragoza, Spain
Email: jguillera@gmail.com

Wadim Zudilin
Affiliation: School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan, NSW 2308, Australia
Email: wadim.zudilin@newcastle.edu.au

DOI: https://doi.org/10.1090/S0002-9939-2011-10950-X
Keywords: Congruence, hypergeometric series, Ramanujan-type identities for $1/\pi$, creative telescoping
Received by editor(s): August 12, 2010
Received by editor(s) in revised form: December 14, 2010
Published electronically: July 21, 2011
Additional Notes: Work of the second author was supported by Australian Research Council grant DP110104419.
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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