A new algorithm for the recursion of multisums with improved universal denominator

TL;DR

This paper introduces two algorithms for hypergeometric multisums: one computes linear recursions with rational certificates, and the other improves the universal denominator method for solving rational recurrence equations.

Contribution

It presents a novel algorithm that enhances Abramov's universal denominator, enabling more efficient computation of rational solutions in hypergeometric multisums.

Findings

The new universal denominator algorithm constructs all rational solutions.

The recursion algorithm provides rational certificates for multisums.

Improved efficiency over previous methods.

Abstract

The purpose of the paper is to introduce two new algorithms. The first one computes a linear recursion for proper hypergeometric multisums, by treating one summation variable at a time, and provides rational certificates along the way. A key part in the search of a linear recursion is an improved universal denominator algorithm that constructs all rational solutions $x(n)$ of the equation $$ \frac{a_m(n)}{b_m(n)}x(n+m)+...+\frac{a_0(n)}{b_0(n)}x(n)= c(n),$$ where $a_i(n), b_i(n), c(n)$ are polynomials. Our algorithm improves Abramov's universal denominator.