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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

Citations:

Zbl 0895.11051
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References:

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