Telescoping method and congruences for double sums

Abstract

In recent years, Z.-W. Sun proposed several sophisticated conjectures on congruences for finite sums with terms involving combinatorial sequences such as central trinomial coefficients, Domb numbers and Franel numbers. These sums are double summations of hypergeometric terms. Using the telescoping method and certain mathematical software packages, we transform such a double summation into a single sum. With this new approach, we confirm several open conjectures of Sun.