On the supercongruence conjectures of van Hamme
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In 1997, van Hamme 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.
KeywordsRamanujan-type supercongruences Hypergeometric series
Mathematics subject classificationPrimary 33C20 Secondary 44A20
The author would like to thank Ling Long for numerous helpful conversations which provided the ideas behind this work, and Robert Osburn for many inspiring conversations and for introducing her to the van Hamme conjectures. The author also thanks Tulane University for hosting her while working on this project, and Sage Days 56, where she ran computations related to this project with Ling Long and Hao Chen.
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