  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.
