“Divergent” Ramanujan-type supercongruences
HTML articles powered by AMS MathViewer
- by Jesús Guillera and Wadim Zudilin PDF
- Proc. Amer. Math. Soc. 140 (2012), 765-777 Request permission
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.References
- Gert Almkvist and Andrew Granville, Borwein and Bradley’s Apéry-like formulae for $\zeta (4n+3)$, Experiment. Math. 8 (1999), no. 2, 197–203. MR 1700578
- J. M. Borwein and P. B. Borwein, More Ramanujan-type series for $1/\pi$, Ramanujan revisited (Urbana-Champaign, Ill., 1987) Academic Press, Boston, MA, 1988, pp. 359–374. MR 938974
- Jonathan Borwein and David Bradley, Empirically determined Apéry-like formulae for $\zeta (4n+3)$, Experiment. Math. 6 (1997), no. 3, 181–194. MR 1481588
- Heng Huat Chan, Song Heng Chan, and Zhiguo Liu, Domb’s numbers and Ramanujan-Sato type series for $1/\pi$, Adv. Math. 186 (2004), no. 2, 396–410. MR 2073912, DOI 10.1016/j.aim.2003.07.012
- Heng Huat Chan and Wadim Zudilin, New representations for Apéry-like sequences, Mathematika 56 (2010), no. 1, 107–117. MR 2604987, DOI 10.1112/S0025579309000436
- Jesús Guillera, Hypergeometric identities for 10 extended Ramanujan-type series, Ramanujan J. 15 (2008), no. 2, 219–234. MR 2377577, DOI 10.1007/s11139-007-9074-0
- Jesús Guillera, A matrix form of Ramanujan-type series for $1/\pi$, Gems in experimental mathematics, Contemp. Math., vol. 517, Amer. Math. Soc., Providence, RI, 2010, pp. 189–206. MR 2731088, DOI 10.1090/conm/517/10141
- J. Guillera, WZ-proofs of “divergent” Ramanujan-type series, preprint at arXiv: 1012.2681 (2010).
- J. Guillera, Mosaic supercongruences of Ramanujan type, preprint at arXiv: 1007.2290 (2010). To appear in Experiment. Math.
- F. Morley, Note on the congruence $2^{4n}\equiv (-)^n(2n)!/(n!)^2$, where $2n+1$ is a prime, Ann. of Math. 9 (1894/95), no. 1-6, 168–170. MR 1502188, DOI 10.2307/1967516
- Marko Petkovšek, Herbert S. Wilf, and Doron Zeilberger, $A=B$, A K Peters, Ltd., Wellesley, MA, 1996. With a foreword by Donald E. Knuth; With a separately available computer disk. MR 1379802
- Lucy Joan Slater, Generalized hypergeometric functions, Cambridge University Press, Cambridge, 1966. MR 0201688
- Tor B. Staver, On summation of powers of binomial coefficients, Norsk Mat. Tidsskr. 29 (1947), 97–103 (Norwegian). MR 24874
- Zhi-Wei Sun, Super congruences and Euler numbers, preprint at arXiv: 1001.4453 (2010).
- Zhi-Wei Sun and Roberto Tauraso, New congruences for central binomial coefficients, Adv. in Appl. Math. 45 (2010), no. 1, 125–148. MR 2628791, DOI 10.1016/j.aam.2010.01.001
- Roberto Tauraso, More congruences for central binomial coefficients, J. Number Theory 130 (2010), no. 12, 2639–2649. MR 2684486, DOI 10.1016/j.jnt.2010.06.004
- V. V. Zudilin, Quadratic transformations and Guillera’s formulas for $1/\pi ^2$, Mat. Zametki 81 (2007), no. 3, 335–340 (Russian, with Russian summary); English transl., Math. Notes 81 (2007), no. 3-4, 297–301. MR 2333939, DOI 10.1134/S0001434607030030
- Wadim Zudilin, Ramanujan-type formulae for $1/\pi$: a second wind?, Modular forms and string duality, Fields Inst. Commun., vol. 54, Amer. Math. Soc., Providence, RI, 2008, pp. 179–188. MR 2454325
- Wadim Zudilin, Ramanujan-type supercongruences, J. Number Theory 129 (2009), no. 8, 1848–1857. MR 2522708, DOI 10.1016/j.jnt.2009.01.013
Similar Articles
- Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11Y55, 33C20, 33F10, 11B65, 11D88, 11F33, 11F85, 11S80, 12H25, 40G99, 65-05, 65B10
- Retrieve articles in all journals with MSC (2010): 11Y55, 33C20, 33F10, 11B65, 11D88, 11F33, 11F85, 11S80, 12H25, 40G99, 65-05, 65B10
Additional Information
- Jesús Guillera
- Affiliation: Av. Cesáreo Alierta, 31 esc. izda 4$^\circ$–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
- 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
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2869062