## On the supercongruence conjectures of van Hamme.(English)Zbl 1337.33005

Summary: In 1997, L. van Hamme [Lect. Notes Pure Appl. Math. 192, 223–236 (1997; Zbl 0895.11051)] developed $$p$$-adic analogs, for primes $$p$$, of several series which relate hypergeometric series to values of the gamma function, originally studied by Ramanujan. These analogs relate truncated sums of hypergeometric series to values of the $$p$$-adic gamma function, and are called Ramanujan-type supercongruences. In all, van Hamme conjectured 13 such formulas, three of which were proved by van Hamme himself, and five others have been proved recently using a wide range of methods. Here, we explore four of the remaining five van Hamme supercongruences, revisit some of the proved ones, and provide some extensions.

### MSC:

 11A07 Congruences; primitive roots; residue systems 11S80 Other analytic theory (analogues of beta and gamma functions, $$p$$-adic integration, etc.) 33C20 Generalized hypergeometric series, $${}_pF_q$$ 33E50 Special functions in characteristic $$p$$ (gamma functions, etc.) 44A20 Integral transforms of special functions

### Keywords:

Ramanujan-type supercongruences; hypergeometric series

Zbl 0895.11051
Full Text:

### References:

 [1] Ahlgren, S; Ono, K, Modularity of a certain Calabi-Yau threefold, Monatshefte für. Mathematik., 129, 177-190, (2000) · Zbl 0999.11031 [2] Andrews, G.E., Askey, R., Roy, R.: Special functions. Encyclopedia of Mathematics and its Applications, vol. 71. Cambridge University Press, Cambridge (1999) [3] Borwein, J.M., Borwein, P.B.: Pi and the AGM. Canadian Mathematical Society Series of Monographs and Advanced Texts. A study in analytic number theory and computational complexity. A Wiley-Interscience Publication, New York (1987) [4] Chisholm, S; Deines, A; Long, L; Nebe, G; Swisher, H, $$p$$-adic analogues of Ramanujan type formulas for 1/$$π$$, Mathematics, 1, 9-30, (2013) · Zbl 1296.11058 [5] Chudnovsky, D.V., Chudnovsky, G.V.: Approximations and complex multiplication according to Ramanujan. In Ramanujan revisited (Urbana-Champaign, Ill., 1987), pp. 375-472. Academic Press, Boston (1988) · Zbl 0647.10002 [6] Dwork, B.: A note on the $$p$$-adic gamma function. In Study group on ultrametric analysis, 9th year: 1981/82, No. 3 (Marseille, 1982), pages Exp. No. J5, 10. Inst. Henri Poincaré, Paris (1983) [7] He, B.: On a $$p$$-adic supercongruence conjecture of l. van hamme. Proc. Amer. Math. Soc. (2015, to appear) · Zbl 1175.33004 [8] He B.: Some congruences on truncated hypergeometric series (2015, preprint) · Zbl 1067.33006 [9] Karlsson, PW, Clausen’s hypergeometric function with variable $$-1/8$$ or $$-8$$, Math. Sci. Res. Hot-Line, 4, 25-33, (2000) · Zbl 1067.33006 [10] Kilbourn, T, An extension of the apéry number supercongruence, Acta. Arith., 123, 335-348, (2006) · Zbl 1170.11008 [11] Long, L, Hypergeometric evaluation identities and supercongruences, Pacific J. Math., 249, 405-418, (2011) · Zbl 1215.33002 [12] Long, L., Ramakrishna, R.: Some supercongruences occurring in truncated hypergeometric series. preprint. http://arxiv.org/abs/1403.5232 · Zbl 1336.33018 [13] McCarthy, D; Osburn, R, A $$p$$-adic analogue of a formula of Ramanujan, Arch. Math. (Basel), 91, 492-504, (2008) · Zbl 1175.33004 [14] Morita, Y, A p-adic analogue of the-function, J. Fac. Sci. Univ. Tokyo, 22, 255-266, (1975) · Zbl 0308.12003 [15] Mortenson, E, A $$p$$-adic supercongruence conjecture of Van hamme, Proc. Amer. Math. Soc., 136, 4321-4328, (2008) · Zbl 1171.11061 [16] Osburn, R., Zudilin, W.: On the (k.2) supercongruence of van hamme (2015, preprint). http://arxiv.org/abs/1504.01976 · Zbl 1400.11062 [17] Rogers, M; Wan, JG; Zucker, IJ, Moments of elliptic integrals and critical $$l$$-values, Ramanujan J., 37, 113-130, (2015) · Zbl 1383.11048 [18] van Hamme, L, Some conjectures concerning partial sums of generalized hypergeometric series, Lect. Notes Pure Appl. Math., 192, 223-236, (1997) · Zbl 0895.11051 [19] Vangeemen, B; Nygaard, NO, On the geometry and arithmetic of some Siegel modular threefolds, J. Number Theor., 53, 45-87, (1995) · Zbl 0838.11047 [20] Verrill, H.A.: Arithmetic of a certain Calabi-Yau threefold. Number theory (Ottawa, ON, 1996), volume 19 of CRM Proc. Lecture Notes, pp. 333-340. Amer. Math. Soc, Providence, RI (1999) · Zbl 0942.14022 [21] Whipple, FJW, On well-poised series, generalized hypergeometric series having parameters in pairs, each pair with the same sum, Proc. London Math. Soc., 24, 247-263, (1926) · JFM 51.0283.03 [22] Zudilin, W, Ramanujan-type supercongruences, J. Number Theor., 129, 1848-1857, (2009) · Zbl 1231.11147
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