## “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.)
Full Text:

### References:

  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  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  Jonathan Borwein and David Bradley, Empirically determined Apéry-like formulae for \?(4\?+3), Experiment. Math. 6 (1997), no. 3, 181 – 194. · Zbl 0887.11037  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  Heng Huat Chan and Wadim Zudilin, New representations for Apéry-like sequences, Mathematika 56 (2010), no. 1, 107 – 117. · Zbl 1275.11035  Jesús Guillera, Hypergeometric identities for 10 extended Ramanujan-type series, Ramanujan J. 15 (2008), no. 2, 219 – 234. · Zbl 1142.33002  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  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$$^{4}$$$$^{n}$$\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  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.  Lucy Joan Slater, Generalized hypergeometric functions, Cambridge University Press, Cambridge, 1966.  Tor B. Staver, On summation of powers of binomial coefficients, Norsk Mat. Tidsskr. 29 (1947), 97 – 103 (Norwegian). · Zbl 0030.28901  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. · Zbl 1231.11021  Roberto Tauraso, More congruences for central binomial coefficients, J. Number Theory 130 (2010), no. 12, 2639 – 2649. · Zbl 1208.11027  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  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  Wadim Zudilin, Ramanujan-type supercongruences, J. Number Theory 129 (2009), no. 8, 1848 – 1857. · Zbl 1231.11147
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.