On the Summability Problem of Multivariate Rational Functions in the Mixed Case

TL;DR

This paper develops criteria for summability of multivariate rational functions involving both shift and q-shift operators, advancing algorithms for symbolic summation in complex multivariate cases.

Contribution

It introduces new summability criteria based on orbital decompositions, Sato's isotropy groups, and difference transformations for mixed shift and q-shift operators.

Findings

Established summability criteria for mixed shift and q-shift cases.

Solved the rational case of a long-term symbolic summation project.

Provided a foundation for future algorithm development in multivariate summation.

Abstract

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.