Continuing previous work, this paper focuses on the summability problem of multivariate rational functions in the mixed case in which both shift and q-shift operators can appear. Our summability criteria rely on three ingredients including orbital decompositions, Sato’s isotropy groups, and difference transformations. This work settles the rational case of the long-term project aimed at developing algorithms for symbolic summation of multivariate functions.

Consider formal power series \(f_1, \ldots , f_k \in \mathbb {Q} [[z]]\) that are defined as the solutions of a system of polynomial differential equations together with a sufficient number of initial conditions. Given \(P \in \mathbb {Q} [F_1, \ldots , F_k]\), several algorithms have been proposed in order to test whether P(f1, …, fk) = 0. In this paper, we present such an algorithm for the case where f1, …, fk are so-called transseries instead of power series.