Using the "Freshman's Dream" to Prove Combinatorial Congruences

TL;DR

This paper explores the use of the 'Freshman's Dream' to prove combinatorial congruences, extending existing algorithms to multivariable Laurent polynomials and multiple sums, with applications to super-congruences.

Contribution

It introduces an extension of an algorithm for proving combinatorial congruences to multivariable Laurent polynomials and multiple sums, enhancing the scope of the original method.

Findings

Extended the algorithm to handle multiple variables.

Applied the method to prove super-congruences.

Confirmed most conjectured super-congruences, with one exception.

Abstract

In a recent beautiful but technical article, William Y.C. Chen, Qing-Hu Hou, and Doron Zeilberger developed an algorithm for finding and proving congruence identities (modulo primes) of indefinite sums of many combinatorial sequences, namely those (like the Catalan and Motzkin sequences) that are expressible in terms of constant terms of powers of Laurent polynomials. We first give a leisurely exposition of their elementary but brilliant approach, and then extend it in two directions. The Laurent polynomials may be of several variables, and instead of single sums we have multiple sums. In fact we even combine these two generalizations! We conclude with some super-challenges. In this version we report that Roberto Tauraso pointed out that all our conjectured super-congruences, at the end of our article are already known, except one, for which he supplied a beautiful proof that can be found here: arXiv:1606.05543.