Research in the Mathematical Sciences· 2015Q2
On the supercongruence conjectures of van Hamme
- 144citations
- Q2SCImago
- 2015year
Short summary
Four of van Hamme's Ramanujan-type supercongruences, which relate truncated hypergeometric series to p-adic gamma function values, are explored and extended.
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Key points
- Explores four of the five remaining van Hamme supercongruence conjectures.
- These conjectures relate truncated hypergeometric series to p-adic gamma function values.
- Revisits some previously proved supercongruences.
- Provides extensions to existing supercongruence results.
AI-generated from the title and abstract; the full text is not read.
Abstract
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.
The authors' abstract, as published at the source. Research in the Mathematical Sciences, 2015 · DOI ↗
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Field: Algebra and Number Theory
Algebra and Number TheoryMathematics