Study on discrete degenerate Bell distributions with two parameters

Recently, Freud-Rodriguez proposed a new counting process which is called the Bell-Touchard process and based on the Bell-Touchard probability distribution. This process was developed to solve the problem of rare events hypothesis which is one of the limitations of the Poisson process. In this paper, we consider the discrete degenerate Bell distributions and the degenerate Bell process which are 'degenerate versions' of the Bell-Touchard probability distributions and the Bell-Touchard process, respectively. We investigate several properties of the degenerate Bell distribution. We introduce the degenerate Bell process by giving two equivalent definitions and show one method of constructing a new infinite family of degenerate Bell process out of a given infinite family of degenerate Bell process.