Discrete inflated data are widely used in practice. One of the most essential approaches for modeling such data involves the use of models based on inflated distributions. Since the choice of the baseline distribution plays a fundamental role in defining a family of inflated distributions, the efficiency and fitting capability of the model are directly related to the baseline distribution. In most research concerning inflated data, models based on the Poisson distribution are employed. The Poisson distribution is a very powerful distribution; however, it possesses one characteristic that becomes its Achilles' heel in application. This characteristic is the equality of the mean and variance—a condition that rarely holds in real-world data. Therefore, it is necessary to seek an alternative to the Poisson distribution, and what better alternative than a broad family of distributions with greater flexibility? One such candidate is the family of Telescopic distributions.‎‎

‎In this research, the family of Telescopic distributions is used as the baseline distribution to define a general class of discrete inflated models. This family includes distributions that have been less frequently used as baseline distributions (such as the discrete Weibull distribution). Moreover, these distributions are also used as lifetime distributions. Additionally, due to their connection with an important family of continuous distributions, they encompass a wide spectrum of other statistical distributions. The breadth of this family is a significant advantage. Thus, if the Telescopic family is used as the baseline distribution for defining inflated models, there will be numerous new models available for fitting any discrete inflated (and even non-inflated) dataset. It is worth noting that this research also briefly addresses a regression model based on the mixing parameter‎.‎‎