Residues and Telescopers for Rational Functions

TL;DR

This paper establishes conditions for the existence of telescopers for rational functions across continuous, discrete, and q-discrete contexts, and characterizes possible operators, extending residue concepts to these settings.

Contribution

It provides a unified framework for residues and telescopers in various settings, rederives classical results, and introduces new characterizations of telescopers for rational functions.

Findings

Necessary and sufficient conditions for telescopers in all settings

Characterization of operators that can serve as telescopers

Generalization of residues to discrete and q-discrete cases

Abstract

We give necessary and sufficient conditions for the existence of telescopers for rational functions of two variables in the continuous, discrete and q-discrete settings and characterize which operators can occur as telescopers. Using this latter characterization, we reprove results of Furstenberg and Zeilberger concerning diagonals of power series representing rational functions. The key concept behind these considerations is a generalization of the notion of residue in the continuous case to an analogous concept in the discrete and q-discrete cases.