Linear birth and death models and associated Laguerre and Meixner polynomials

Abstract

We study birth and death processes with linear rates λ n = n + α + c + 1, μ n + 1 = n + c, n ⩾ 0 and μ 0 is either zero or c. The spectral measures of both processes are found using generating functions and the integral transforms of Laplace and Stieltjes. The corresponding orthogonal polynomials generalize Laguerre polynomials and the choice μ 0 = c generates the associated Laguerre polynomials of Askey and Wimp. We investigate the orthogonal polynomials in both cases and give alternate proofs of some of the results of Askey and Wimp on the associated Laguerre polynomials. We also identify the spectra of the associated Charlier and Meixner polynomials as zeros of certain transcendental equations.