Bivariate Extensions of Abramov's Algorithm for Rational Summation

TL;DR

This paper completes the extension of Abramov's algorithm to all bivariate cases, enabling rational summation in mixed discrete variables, building on prior univariate and specific bivariate cases.

Contribution

It solves the remaining three mixed cases of bivariate Abramov's algorithm, completing the framework for rational summation in two discrete variables.

Findings

Complete solution for all mixed bivariate cases

Extension of Abramov's algorithm to new variable configurations

Facilitates rational summation in complex bivariate scenarios

Abstract

Abramov's algorithm enables us to decide whether a univariate rational function can be written as a difference of another rational function, which has been a fundamental algorithm for rational summation. In 2014, Chen and Singer generalized Abramov's algorithm to the case of rational functions in two ($q$-)discrete variables. In this paper we solve the remaining three mixed cases, which completes our recent project on bivariate extensions of Abramov's algorithm for rational summation.