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

Guillera, Jesús; Zudilin, Wadim

Proc. Am. Math. Soc. 140, No. 3, 765-777 (2012).

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)].

Reviewer: Enzo Bonacci (Latina)

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.)

Keywords:

congruences; Ramanujan-type identities; Wilf-Zeilberger pairs; hypergeometric series

Citations:

Zbl 1142.33002; Zbl 1285.11151; Zbl 1207.33012; Zbl 1247.11153; Zbl 1144.33002; Zbl 1256.11011; Zbl 1159.11053; Zbl 1231.11147; Zbl 0848.05002; Zbl 0030.28901; Zbl 0887.11037; Zbl 0976.11035; Zbl 0135.28101; Zbl 1231.11021; Zbl 0652.10019; Zbl 1122.11087; Zbl 1275.11035; JFM 26.0208.02

References:

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