Supercongruences involving Domb numbers and binary quadratic forms

Abstract

In this paper, we prove two recently conjectured supercongruences (modulo $p^3$, where p is any prime greater than 3) of Zhi-Hong Sun on truncated sums involving the Domb numbers. Our proofs involve a number of ingredients such as congruences involving specialized Bernoulli polynomials, harmonic numbers, binomial coefficients, and hypergeometric summations and transformations.

Keywords: Binary quadratic forms; Binomial coefficients; Congruences; Domb numbers.