Proof of some conjectural hypergeometric supercongruences via curious identities

Gaussian Hypergeometric series and supercongruences

Let p be an odd prime. In 1984, Greene introduced the notion of hypergeometric functions over finite fields. Special values of these functions have been of interest as they are related to the number of Fp points on algebraic varieties and to Fourier coefficients of modular forms. In this paper, we explicitly determine these functions modulo higher powers of p and discuss an application to super...

Gaussian Hypergeometric Series and Extensions of Supercongruences

Let p be an odd prime. The purpose of this paper is to refine methods of Ahlgren and Ono [2] and Kilbourn [13] in order to prove a general mod p congruence for the Gaussian hypergeometric series n+1Fn(λ) where n is an odd positive integer. As a result, we extend three recent supercongruences. The first is a result of Ono and Ahlgren [2] on a supercongruence for Apéry numbers which was conjectur...

Supercongruences via Modular Forms

We prove two supercongruences for the coefficients of power series expansions in t of modular forms where t is a modular function. As a result, we settle two recent conjectures of Chan, Cooper and Sica. Additionally, we provide a table of supercongruences for numbers which appear in similar power series expansions and in the study of integral solutions of Apéry-like differential equations.

A New Proof of Some Identities of Bressoud

Binomial Coefficient–harmonic Sum Identities Associated to Supercongruences

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 > 1) congruences between truncated generalized hypergeometric series, and a function which extends Greene’s ...