  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.
