Bivariate Extensions of Abramov's Algorithm for Rational Summation

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.