Exactly solvable discrete time Birth and Death processes

TL;DR

This paper provides 15 explicit examples of exactly solvable discrete-time Birth and Death processes linked to hypergeometric orthogonal polynomials, detailing their rates, distributions, and transition probabilities.

Contribution

It introduces a set of discrete-time models explicitly solvable using orthogonal polynomials from the Askey scheme, extending continuous-time solutions to discrete time.

Findings

Explicit birth and death rates derived from polynomial difference equations

Stationary distributions correspond to orthogonality measures of polynomials

Transition probabilities expressed via normalized polynomials and eigenvalues

Abstract

We present 15 explicit examples of discrete time Birth and Death processes which are exactly solvable. They are related to the hypergeometric orthogonal polynomials of 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 the difference equations governing the polynomials. The stationary distributions are the normalised orthogonality measures of the polynomials. The transition probabilities are neatly expressed by the normalised 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.