On the 2-orthogonal polynomials and the generalized birth and death processes

The birth and death processes are closely related to the orthogonal polynomials. The latter allows determining the stochastic matrix associated with these processes. Let us also note that these processes are stationary Markov processes whose state space is the nonnegative integers. Many authors treated the question of the existing relationship between the birth and death processes and the orthogonal polynomials, in particular, in the works of Karlin and McGregor [6] and Ismail et al. [5]. The properties of these processes and of the orthogonal polynomials were the subject of other works, we can quote by the way of example Ismail et al. [3, 4], Maki [8], and Letessier and Valent [7]. In this paper, we will consider not only the orthogonal polynomials, but the 2-orthogonal polynomials and we will try to establish a bond between the latter and certain birth and death processes that will be called “generalized.” These processes will have like transition probabilities