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“Divergent” Ramanujan-type supercongruences. (English) Zbl 1276.11027

Each of the authors had individually investigated Ramanujan-type formulae (J. Guillera [Ramanujan J. 15, No. 2, 219–234 (2008; Zbl 1142.33002)]; [Contemp. Math. 517, 189–206 (2010; Zbl 1207.33012)]; [“WZ-proofs of “divergent” Ramanujan-type series”, Advances in combinatorics. Waterloo, Canada, 2011. Berlin: Springer, 187–195 (2013; Zbl 1285.11151), see also arXiv:1012.2681]; [Exp. Math. 21, No. 1, 65–68 (2012; Zbl 1247.11153)] and W. Zudilin [Math. Notes 81, No. 3, 297–301 (2007); translation from Mat. Zametki 81, No. 3, 335–340 (2007; Zbl 1144.33002)], [Modular forms and string duality. Proceedings of a workshop, Banff, Canada, 2006. Fields Inst. Commun. 54, 179-188 (2008; Zbl 1159.11053)]) hence this paper extends and refines various previous works, especially by W. Zudilin [J. Number Theory 129, No. 8, 1848–1857 (2009; Zbl 1231.11147)].
Here the authors establish the following supercongruences related to “divergent” Ramanujan-type series for \( 1/\pi\) and \( 1/\pi^2\):
\[ \sum_{n=0}^{(p-1)/2}\frac{(\frac 1 2)^3_n}{{n!}^3} (3n+1)2^{2n} \equiv p \pmod{p^3} \text{ for primes } p>2, \]
\[ \sum_{n=0}^{(p-1)/2}\frac{(\frac 1 2)^5_n}{{n!}^3}(10n^2+6n+1) (-1)^n 2^{2n} \equiv {p^2} \pmod{p^5} \text{ for primes } p>3, \]
\[ \sum_{k=0}^{p-1}\frac{(\frac 1 2)^3_n}{(1)^3_n} (3n+1) (-1)^n 2^{3n} \equiv(-1)^{(p-1)/2}p \pmod{p^3} \text{ for primes } p>2. \] Although they find other supercongruences of the same kind, the authors remark that most of them have been independently discovered by Zhi-Wei Sun [“Super congruences and Euler numbers”, Sci. China, Math. 54, No. 12, 2509–2535 (2011; Zbl 1256.11011)].
Beyond a versatile use of the \(WZ\)-pairs algorithmic technique by M. Petkovšek, H. S. Wilf and D. Zeilberger [\(A=B\). Wellesley, MA: A. K. Peters (1996; Zbl 0848.05002)] in the proof the authors employ an identity by T. B. Staver [Norsk Mat. Tidsskr. 29, 97–103 (1947; Zbl 0030.28901)], a congruence by R. Tauraso [“Congruences involving the reciprocals of central binomial coefficients”, arXiv:0906.5150], a conjecture formulated by J. Borwein and D. Bradley [Exp. Math. 6, No.3, 181-194 (1997; Zbl 0887.11037)] and proved by G. Almkvist and A. Granville [Exp. Math. 8, No. 2, 197–203 (1999; Zbl 0976.11035)], the Chu-Vandermonde theorem studied by [L. J. Slater, Generalized hypergeometric functions. Cambridge: At the University Press (1966; Zbl 0135.28101)], the congruence supplied by F. Morley [Ann. Math. 9, 168–170 (1895; JFM 26.0208.02)], a result from Z.-W. Sun and R. Tauraso [Adv. Appl. Math. 45, No. 1, 125–148 (2010; Zbl 1231.11021)], a method given by H. H. Chan and W. Zudilin [Mathematika 56, No. 1, 107–117 (2010; Zbl 1275.11035)] and the general machinery for proving Ramanujan-like series for \( 1/\pi\) developed by J. M. Borwein and P. B. Borwein [Ramanujan revisited, Proc. Conf., Urbana-Champaign/Illinois 1987, 359–374 (1988; Zbl 0652.10019)] and by H. H. Chan, S. H. Chan and Z. Liu [Adv. Math. 186, No. 2, 396–410 (2004; Zbl 1122.11087)].

MSC:

11B65 Binomial coefficients; factorials; \(q\)-identities
11A07 Congruences; primitive roots; residue systems
33C20 Generalized hypergeometric series, \({}_pF_q\)
33F10 Symbolic computation of special functions (Gosper and Zeilberger algorithms, etc.)
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References:

[1] Gert Almkvist and Andrew Granville, Borwein and Bradley’s Apéry-like formulae for \?(4\?+3), Experiment. Math. 8 (1999), no. 2, 197 – 203. · Zbl 0976.11035
[2] J. M. Borwein and P. B. Borwein, More Ramanujan-type series for 1/\?, Ramanujan revisited (Urbana-Champaign, Ill., 1987) Academic Press, Boston, MA, 1988, pp. 359 – 374. · Zbl 0652.10019
[3] Jonathan Borwein and David Bradley, Empirically determined Apéry-like formulae for \?(4\?+3), Experiment. Math. 6 (1997), no. 3, 181 – 194. · Zbl 0887.11037
[4] Heng Huat Chan, Song Heng Chan, and Zhiguo Liu, Domb’s numbers and Ramanujan-Sato type series for 1/\?, Adv. Math. 186 (2004), no. 2, 396 – 410. · Zbl 1122.11087
[5] Heng Huat Chan and Wadim Zudilin, New representations for Apéry-like sequences, Mathematika 56 (2010), no. 1, 107 – 117. · Zbl 1275.11035
[6] Jesús Guillera, Hypergeometric identities for 10 extended Ramanujan-type series, Ramanujan J. 15 (2008), no. 2, 219 – 234. · Zbl 1142.33002
[7] Jesús Guillera, A matrix form of Ramanujan-type series for 1/\?, Gems in experimental mathematics, Contemp. Math., vol. 517, Amer. Math. Soc., Providence, RI, 2010, pp. 189 – 206. · Zbl 1207.33012
[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\(^{4}\)\(^{n}\)\equiv (-)\(^{n}\)(2\?)!/(\?!)², where 2\?+1 is a prime, Ann. of Math. 9 (1894/95), no. 1-6, 168 – 170. · JFM 26.0208.02
[11] Marko Petkovšek, Herbert S. Wilf, and Doron Zeilberger, \?=\?, A K Peters, Ltd., Wellesley, MA, 1996. With a foreword by Donald E. Knuth; With a separately available computer disk.
[12] Lucy Joan Slater, Generalized hypergeometric functions, Cambridge University Press, Cambridge, 1966.
[13] Tor B. Staver, On summation of powers of binomial coefficients, Norsk Mat. Tidsskr. 29 (1947), 97 – 103 (Norwegian). · Zbl 0030.28901
[14] ZHI-WEI SUN, Super congruences and Euler numbers, preprint at arXiv: 1001.4453 (2010).
[15] Zhi-Wei Sun and Roberto Tauraso, New congruences for central binomial coefficients, Adv. in Appl. Math. 45 (2010), no. 1, 125 – 148. · Zbl 1231.11021
[16] Roberto Tauraso, More congruences for central binomial coefficients, J. Number Theory 130 (2010), no. 12, 2639 – 2649. · Zbl 1208.11027
[17] V. V. Zudilin, Quadratic transformations and Guillera’s formulas for 1/\?², Mat. Zametki 81 (2007), no. 3, 335 – 340 (Russian, with Russian summary); English transl., Math. Notes 81 (2007), no. 3-4, 297 – 301. · Zbl 1144.33002
[18] Wadim Zudilin, Ramanujan-type formulae for 1/\?: a second wind?, Modular forms and string duality, Fields Inst. Commun., vol. 54, Amer. Math. Soc., Providence, RI, 2008, pp. 179 – 188. · Zbl 1159.11053
[19] Wadim Zudilin, Ramanujan-type supercongruences, J. Number Theory 129 (2009), no. 8, 1848 – 1857. · Zbl 1231.11147
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