Supercongruences and truncated hypergeometric series

He, Bing

doi:10.1090/proc/13305

Supercongruences and truncated hypergeometric series
HTML articles powered by AMS MathViewer

by Bing He PDF

Proc. Amer. Math. Soc. 145 (2017), 501-508 Request permission

Abstract:

We employ the $p$-adic Gamma function and some formulas on hypergeometric series to establish several new supercongruences.

References

George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958, DOI 10.1017/CBO9781107325937
Victor V. Batyrev and Duco van Straten, Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections in toric varieties, Comm. Math. Phys. 168 (1995), no. 3, 493–533. MR 1328251
Heng Huat Chan, Shaun Cooper, and Francesco Sica, Congruences satisfied by Apéry-like numbers, Int. J. Number Theory 6 (2010), no. 1, 89–97. MR 2641716, DOI 10.1142/S1793042110002879
Henri Cohen, Number theory. Vol. II. Analytic and modern tools, Graduate Texts in Mathematics, vol. 240, Springer, New York, 2007. MR 2312338
B. Dwork, $p$-adic cycles, Inst. Hautes Études Sci. Publ. Math. 37 (1969), 27–115. MR 294346
Ira Gessel and Dennis Stanton, Strange evaluations of hypergeometric series, SIAM J. Math. Anal. 13 (1982), no. 2, 295–308. MR 647127, DOI 10.1137/0513021
Timothy Kilbourn, An extension of the Apéry number supercongruence, Acta Arith. 123 (2006), no. 4, 335–348. MR 2262248, DOI 10.4064/aa123-4-3
Ling Long, Hypergeometric evaluation identities and supercongruences, Pacific J. Math. 249 (2011), no. 2, 405–418. MR 2782677, DOI 10.2140/pjm.2011.249.405
Ling Long and Ravi Ramakrishna, Some supercongruences occurring in truncated hypergeometric series, Adv. Math. 290 (2016), 773–808. MR 3451938, DOI 10.1016/j.aim.2015.11.043
Dermot McCarthy and Robert Osburn, A $p$-adic analogue of a formula of Ramanujan, Arch. Math. (Basel) 91 (2008), no. 6, 492–504. MR 2465868, DOI 10.1007/s00013-008-2828-0
Eric Mortenson, A $p$-adic supercongruence conjecture of van Hamme, Proc. Amer. Math. Soc. 136 (2008), no. 12, 4321–4328. MR 2431046, DOI 10.1090/S0002-9939-08-09389-1
Eric Mortenson, Supercongruences between truncated ${}_2F_1$ hypergeometric functions and their Gaussian analogs, Trans. Amer. Math. Soc. 355 (2003), no. 3, 987–1007. MR 1938742, DOI 10.1090/S0002-9947-02-03172-0
Eric Mortenson, Supercongruences for truncated $_{n+1}\!F_n$ hypergeometric series with applications to certain weight three newforms, Proc. Amer. Math. Soc. 133 (2005), no. 2, 321–330. MR 2093051, DOI 10.1090/S0002-9939-04-07697-X
Ken Ono, The web of modularity: arithmetic of the coefficients of modular forms and $q$-series, CBMS Regional Conference Series in Mathematics, vol. 102, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2004. MR 2020489
Zhi-Hong Sun, Congruences concerning Legendre polynomials, Proc. Amer. Math. Soc. 139 (2011), no. 6, 1915–1929. MR 2775368, DOI 10.1090/S0002-9939-2010-10566-X
Zhi-Hong Sun, Congruences concerning Legendre polynomials II, J. Number Theory 133 (2013), no. 6, 1950–1976. MR 3027947, DOI 10.1016/j.jnt.2012.11.004
Zhi-Hong Sun, Congruences concerning Legendre polynomials III, Int. J. Number Theory 9 (2013), no. 4, 965–999. MR 3060870, DOI 10.1142/S1793042113500097
Zhi-Hong Sun, Generalized Legendre polynomials and related supercongruences, J. Number Theory 143 (2014), 293–319. MR 3227350, DOI 10.1016/j.jnt.2014.04.012
Z.-W. Sun, Open conjectures on congruences, arxiv.org/abs/0911.5665
Zhi-Wei Sun, Super congruences and Euler numbers, Sci. China Math. 54 (2011), no. 12, 2509–2535. MR 2861289, DOI 10.1007/s11425-011-4302-x
L. van Hamme, Some conjectures concerning partial sums of generalized hypergeometric series, $p$-adic functional analysis (Nijmegen, 1996) Lecture Notes in Pure and Appl. Math., vol. 192, Dekker, New York, 1997, pp. 223–236. MR 1459212
Chuanan Wei, Dianxuan Gong, and Jianbo Li, $\pi$-formulas with free parameters, J. Math. Anal. Appl. 396 (2012), no. 2, 880–887. MR 2961278, DOI 10.1016/j.jmaa.2012.07.039
Wadim Zudilin, Ramanujan-type supercongruences, J. Number Theory 129 (2009), no. 8, 1848–1857. MR 2522708, DOI 10.1016/j.jnt.2009.01.013