Darboux transformations of linear birth-death processes

Abstract

In a previous paper we characterized the lower-upper (LU) and upper-lower (UL) factorizations of the infinitesimal generator of a possibly nonconservative birth-death process whose Darboux transformations are again generators of birth-death processes. Here we apply these results to several classical models with linear rates and explicitly known spectral measures: the $M/M/\infty$ queue (Charlier polynomials), the linear birth-death processes (Meixner and Laguerre polynomials), and the linear models of Ismail, Letessier and Valent (associated Laguerre polynomials). For each of them we compute the factors in closed form, determine the admissible ranges of the parameters, and describe the rates, spectral measures and orthogonal polynomials of the Darboux-transformed processes.