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Binomial coefficient-harmonic sum identities associated to supercongruences. (English) Zbl 1234.05039

Summary: 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 \(p^{k}\) \((k > 1)\) 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.

MSC:

11B65 Binomial coefficients; factorials; \(q\)-identities
05A19 Combinatorial identities, bijective combinatorics
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