Exactly solvable discrete time birth and death processes

Abstract

We present 15 explicit examples of discrete time birth and death processes which are exactly solvable. They are related to hypergeometric orthogonal polynomials of the Askey scheme having discrete orthogonality measures. Namely, they are the Krawtchouk, three different kinds of q-Krawtchouk, (dual, q)-Hahn, (q)-Racah, Al-Salam–Carlitz II, q-Meixner, q-Charlier, dual big q-Jacobi, and dual big q-Laguerre polynomials. The birth and death rates are determined by using the difference equations governing the polynomials. The stationary distributions are the normalized orthogonality measures of the polynomials. The transition probabilities are neatly expressed by the normalized polynomials and the corresponding eigenvalues. This paper is simply the discrete time versions of the known solutions of the continuous time birth and death processes.