Residues and telescopers for bivariate rational functions

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.

Walks in the quarter plane: Genus zero case

“…This problem is known as a q-summation problem and has been solved by Abramov [Abr95]. This procedure was recast in [CS12] in terms of the so-called q-residues of b which we now define.…”

Bivariate Extensions of Abramov’s Algorithm for Rational Summation

“…Assume that d i = w j 1 . If φ (w j 1 ) = w j 1 , then w j 1 ∈ k[y] by [15,Lemma 3.4]. Otherwise, φ (w j 1 ) = w j 2 for some j 2 ∈ {1, .…”

Some open problems related to creative telescoping

“…In a way, this would be the inverse problem of creative telescoping. Chen and Singer in [31] gave a characterization of possible linear operator that can be minimal telescopers for bivariate rational functions. However, no algorithm is known for solving this problem in the general case, but it would be very valuable for practical applications.…”