A p-adic analogue of a formula of Ramanujan

By Dermot McCarthy, Robert Osburn. 2007

81 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/b221d9998020b867670fbc1236a9bf94ba3f4c92

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In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

Citations in the corpus

On the (K.2) supercongruence of Van Hamme (2015, background): “2), while McCarthy and the first author [8] combined this approach with Whipple’s transformation to prove (A.”
Some congruences related to a congruence of Van Hamme (2019, background): “Motivated byMcCarthy and Osburn [33] andMortenson [3], we take the following choice of variables in (3.1).”
On Van Hamme's (A.2) and (H.2) supercongruences (2018, background): “[8] R. Osburn and W. Zudilin, On the (K.2) supercongruence of Van Hamme, J. Math.”
Super congruences and Euler numbers (2011, methodology): “pX−1 k=0 2k k 2 16k ≡ (−1)(p−1)/2 (mod p2), which was later proved by E. Mortenson [M03a] via the p-adic Γ-function and the Gross-Koblitz formula. See also K. Ono’s book [O] and the papers [AO], [K], [MO], [M03b], [M05], [M08], [Mc] and [OS] for such advanced approach to super congruences. Recently ”
On a supercongruence conjecture of Rodriguez-Villegas (2012, methodology): “The main approach for proving these types of supercongruences has been to use congruence relations between truncated generalized hypergeometric series and Greene’s hypergeometric functions over finite fields [1, 2, 11, 16, 18, 19, 20].”
Super congruences and Euler numbers (2010, methodology): “pX−1 k=0 2k k 2 16k ≡ (−1)(p−1)/2 (mod p2), which was later proved by E. Mortenson [M03a] via the p-adic Γ-function and the Gross-Koblitz formula. See also K. Ono’s book [O] and the papers [AO], [K], [MO], [M03b], [M05], [M08], [Mc] and [OS] for such advanced approach to super congruences. Recently ”
A q-microscope for supercongruences (2018, methodology): “Development of methods [7, 20–22, 24, 25, 27, 31, 32, 34] for establishing Ramanujan-type supercongruences like (2) and (3), sometimes modulo a smaller power of p and normally on a case-by-case study, was mainly hypergeometric.”
Ramanujan-type supercongruences (2008, methodology): “We have at least two successful examples of their application: D. McCarthy and R. Osburn [ 7 ] use them to prove a different supercongruence conjectured in [12], and E. Mortenson [10] gives a similar proof of the supercongruence (2).”