On the first meeting or crossing of two independent trajectories for some counting processes

The paper is concerned with the first meeting or crossing problem between two independent trajectories for some basic counting processes. Our interest is focused on the exact distribution of the level and the time of this first meeting or crossing. The question is examined for a renewal process with successively a compound Poisson process, a compound binomial process or a linear birth process with immigration. For each case, a separate analysis is made according as the trajectory of the renewal process starts under or above the trajectory of the other process. A general and systematic approach is developed that uses, as a mathematical tool, a randomized version of two families of polynomials of Abel-Gontcharoff and Appell types.