Autoregressive models adapted to count time series have received less attention than their classical counterparts for continuous time series. The main approach is based on thinning stochastic operation that preserves the discrete nature of the variable between successive times. The binomial thinning operator is the most popular and the Poisson distribution emerges as the natural choice for the error distribution of such an autoregressive counting process. The present paper introduces the quasi Pólya thinning operator, that includes the binomial thinning operator as a special case. The family of additive modified power series distribution is defined and is shown to be the natural choice for the error distribution of such a counting process. We obtain the most general class of integer-valued autoregressive models of order 1 (INAR(1) models) with margins having analytic form and the property of closure under convolution introduced by Joe (1996). It includes the usual cases of Poisson and generalized Poisson margins, but also the less usual cases of binomial and negative binomial margins and the new case of generalized negative binomial margin. These models cover a high range of dispersion that are strictly ordered from the binomial case to the generalized negative binomial case. Asymptotic normality of the maximum likelihood estimator (MLE) for such INAR(1) models is obtained. Finally, the proposed INAR(1) models are applied on simulated and real datasets.