``Divergent'' Ramanujan-type supercongruences (Q2880637)

Each of the authors had individually investigated Ramanujan-type formulae (\textit{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 \url{arXiv:1012.2681}]; [Exp. Math. 21, No. 1, 65--68 (2012; Zbl 1247.11153)] and \textit{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 \textit{W. Zudilin} [J. Number Theory 129, No. 8, 1848--1857 (2009; Zbl 1231.11147)].NEWLINENEWLINEHere the authors establish the following supercongruences related to ``divergent'' Ramanujan-type series for \( 1/\pi\) and \( 1/\pi^2\): NEWLINENEWLINE\[NEWLINE\sum_{n=0}^{(p-1)/2}\frac{(\frac 1 2)^3_n}{{n!}^3} (3n+1)2^{2n} \equiv p \pmod{p^3} \text{ for primes } p>2,NEWLINE\]NEWLINE NEWLINENEWLINE\[NEWLINE\sum_{n=0}^{(p-1)/2}\frac{(\frac 1 2)^5_n}{{n!}^3}(10n^2+6n+1) (-1)^n 2^{2n} \equiv {p^2} \pmod{p^5} \text{ for primes } p>3,NEWLINE\]NEWLINE NEWLINENEWLINE\[NEWLINE \sum_{k=0}^{p-1}\frac{(\frac 1 2)^3_n}{(1)^3_n} (3n+1) (-1)^n 2^{3n} \equiv(-1)^{(p-1)/2}p \pmod{p^3} \text{ for primes } p>2.NEWLINE\]NEWLINE Although they find other supercongruences of the same kind, the authors remark that most of them have been independently discovered by \textit{Zhi-Wei Sun} [``Super congruences and Euler numbers'', Sci. China, Math. 54, No. 12, 2509--2535 (2011; Zbl 1256.11011)].NEWLINENEWLINEBeyond a versatile use of the \(WZ\)-pairs algorithmic technique by \textit{M. Petkovšek, H. S. Wilf} and \textit{D. Zeilberger} [\(A=B\). Wellesley, MA: A. K. Peters (1996; Zbl 0848.05002)] in the proof the authors employ an identity by \textit{T. B. Staver} [Norsk Mat. Tidsskr. 29, 97--103 (1947; Zbl 0030.28901)], a congruence by \textit{R. Tauraso} [``Congruences involving the reciprocals of central binomial coefficients'', \url{arXiv:0906.5150}], a conjecture formulated by \textit{J. Borwein} and \textit{D. Bradley} [Exp. Math. 6, No.3, 181-194 (1997; Zbl 0887.11037)] and proved by \textit{G. Almkvist} and \textit{A. Granville} [Exp. Math. 8, No. 2, 197--203 (1999; Zbl 0976.11035)], the Chu-Vandermonde theorem studied by [\textit{L. J. Slater}, Generalized hypergeometric functions. Cambridge: At the University Press (1966; Zbl 0135.28101)], the congruence supplied by \textit{F. Morley} [Ann. Math. 9, 168--170 (1895; JFM 26.0208.02)], a result from \textit{Z.-W. Sun} and \textit{R. Tauraso} [Adv. Appl. Math. 45, No. 1, 125--148 (2010; Zbl 1231.11021)], a method given by \textit{H. H. Chan} and \textit{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 \textit{J. M. Borwein} and \textit{P. B. Borwein} [Ramanujan revisited, Proc. Conf., Urbana-Champaign/Illinois 1987, 359--374 (1988; Zbl 0652.10019)] and by \textit{H. H. Chan, S. H. Chan} and \textit{Z. Liu} [Adv. Math. 186, No. 2, 396--410 (2004; Zbl 1122.11087)].