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

2026-02-03

AI Summary

This study addresses the problem of determining the summability of multivariate rational functions involving both shift and q-shift operators simultaneously. By integrating orbit decomposition, Sato’s theory of isotropy groups, and differential transformation techniques, the authors establish—for the first time—a complete summability criterion within a mixed difference operator framework. This work fills a critical theoretical gap concerning symbolic summation of multivariate rational functions in mixed shift settings and provides a foundational basis for the systematic development of symbolic summation algorithms.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text

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.

Problem

• summability
• multivariate rational functions
• mixed case
• shift operators
• q-shift operators

Innovation

• summability
• multivariate rational functions
• mixed case
• orbital decompositions
• Sato's isotropy groups