  We introduce a two-parameter expectation thinning operator based on a linear
fractional probability generating function. The operator is then used to define
a first-order integer-valued autoregressive \inar1 process. Distributional
properties of the \inar1 process are described. We revisit the
Bernoulli-geometric \inar1 process of Bourguignon and Wei{\ss} (2017) and we
introduce a new stationary \inar1 process with a compound negative binomial
distribution. Lastly, we show how a proper randomization of our operator leads
to a generalized notion of monotonicity for distributions on \bzp.
