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Binomial Coefficient – Harmonic Sum Identities Associated to Supercongruences

Dermot McCarthy
Published Online: 2011-08-04 | DOI: https://doi.org/10.1515/INTEG.2011.061


We establish two binomial coefficient-generalized harmonic sum identities using the partial fraction decomposition method. These identities are a key ingredient in the proofs of numerous supercongruences. In particular, in other works of the author, they are used to establish modulo pk (k > 1) congruences between truncated generalized hypergeometric series, and a function which extends Greene's hypergeometric function over finite fields to the p-adic setting. A specialization of one of these congruences is used to prove an outstanding conjecture of Rodriguez-Villegas which relates a truncated generalized hypergeometric series to the p-th Fourier coefficient of a particular modular form.

Keywords.: Binomial Coefficient; Harmonic Sum; Identity; Supercongruence; Hypergeometric Function

About the article

Received: 2011-01-31

Accepted: 2011-05-03

Published Online: 2011-08-04

Published in Print: 2011-12-01

Citation Information: Integers, Volume 11, Issue 6, Pages 801–809, ISSN (Print) 1867-0652, DOI: https://doi.org/10.1515/INTEG.2011.061.

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