A p-adic analogue of a formula of Ramanujan

Abstract

During his lifetime, Ramanujan provided many formulae relating binomial sums to special values of the Gamma function. Based on numerical computations, Van Hamme recently conjectured $p$-adic analogues to such formulae. Using a combination of ordinary and Gaussian hypergeometric series, we prove one of these conjectures.

10 pages

Cited by in corpus (22)

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• Some new $q$-congruences for truncated basic hypergeometric series: even powers
• Some q-analogues of supercongruences of Rodriguez-Villegas
• Supercongruences for sporadic sequences
• Extending Gaussian hypergeometric series to the $p$-adic setting
• A $p$-adic supercongruence for truncated hypergeometric series ${}_7{F}_6$
• Some supercongruences occurring in truncated hypergeometric series
• Distribution of values of Gaussian hypergeometric functions
• Supercongruences for truncated hypergeometric series and p-adic gamma function
• On a supercongruence conjecture of Rodriguez-Villegas
• Some q-analogues of (super)congruences of Beukers, Van Hamme and Rodriguez-Villegas
• Some congruences related to a congruence of Van Hamme
• Some $q$-supercongruences modulo the fourth power of a cyclotomic polynomial
• Hypergeometric evaluation identities and supercongruences
• Streamlined WZ method proofs of Van Hamme supercongruences
• Some q-supercongruences from Rahman's summation formula
• Supercongruences arising from hypergeometric series identities
• On van Hamme's (A.2) and (H.2) supercongruences