On the supercongruence conjectures of van Hamme

In 1997, van Hamme developed $$p$$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 $$p$$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.

[1]  B. Dwork.  A note on the $p$-adic gamma function , 1982 .

[2]  N. Nygaard,et al.  On the Geometry and Arithmetic of Some Siegel Modular Threefolds , 1995 .

[3]  Jonathan M. Borwein,et al.  Pi and the AGM , 1999 .

[4]  T. Kilbourn.  An extension of the Apéry number supercongruence , 2006 .

[5]  R. Osburn,et al.  A p-adic analogue of a formula of Ramanujan , 2007, 0708.3307.

[6]  Bing He,et al.  Supercongruences on Truncated Hypergeometric Series , 2015 .

[7]  Leon M. Hall,et al.  Special Functions , 1998 .

[8]  D. V. Chudnovsky,et al.  Approximations and complex multiplication according to Ramanujan , 2000 .

[9]  G. Nebe,et al.  ρ — Adic Analogues of Ramanujan Type Formulas for 1/π , 2013 .

[10]  L. Long,et al.  Some supercongruences occurring in truncated hypergeometric series , 2014, 1403.5232.

[11]  Wadim Zudilin,et al.  Ramanujan-type supercongruences , 2008, 0805.2788.

[12]  Frits Beukers,et al.  SPECIAL FUNCTIONS (Encyclopedia of Mathematics and its Applications 71) , 2001 .

[13]  Eric T. Mortenson,et al.  A p-ADIC SUPERCONGRUENCE CONJECTURE OF VAN HAMME , 2008 .

[14]  Ling Long,et al.  Hypergeometric evaluation identities and supercongruences , 2009, 0912.0197.

[15]  H. Verrill.  Arithmetic of a certain Calabi-Yau threefold , 1999 .

[16]  Yasuo Morita,et al.  A p-adic analogue of the $\Gamma$-function , 1975 .

[17]  G. Andrews,et al.  Encyclopedia of Mathematics and its Applications , 1990 .

[18]  I. Zucker,et al.  Moments of elliptic integrals and critical $$L$$L-values , 2013, 1303.2259.

[19]  F. J. W. Whipple,et al.  On Well-Poised Series, Generalized Hypergeometric Series having Parameters in Pairs, each Pair with the Same Sum , 1926 .

[20]  R. Osburn,et al.  On the (K.2) supercongruence of Van Hamme , 2015, 1504.01976.