Telescoping method and congruences for double sums

TL;DR

This paper introduces a telescoping method combined with software tools to transform complex double sums involving combinatorial sequences into simpler single sums, confirming several of Sun's conjectures on congruences.

Contribution

It presents a novel telescoping approach and computational technique to prove conjectures on congruences for sums involving combinatorial sequences.

Findings

Confirmed several of Sun's conjectures on congruences

Transformed double sums into single sums using telescoping

Validated the effectiveness of the method with computational tools

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.