2-orthogonal polynomials and linear “2–generalized” birth and death processes

Abstract

We show that the linear ”2-generalized” birth and death processes (i.e. the processes defined from two death rates and one birth rate) could be generated by the 2-orthogonal polynomials sets. In particular, we give a characterization of these processes when their related 2-orthogonal polynomials are Sheffer-Meixner type. Also, we show that in one particular case, it is possible to give an integral representation of the measures of orthogonality. In the general case, we give the integral equations satisfied by the measures of orthogonality.