Exactly Solvable Birth and Death Processes

TL;DR

This paper presents a collection of exactly solvable birth and death processes with explicit transition probabilities, derived through a novel connection to matrix quantum mechanics and hypergeometric orthogonal polynomials.

Contribution

It introduces a unified framework for exactly solvable birth and death processes using similarity transformations of matrix quantum mechanics and hypergeometric polynomials.

Findings

Explicit transition probabilities for various birth-death processes

Connection between solvable processes and hypergeometric orthogonal polynomials

Identification of the most general solvable rates as rational functions of q^x

Abstract

Many examples of exactly solvable birth and death processes, a typical stationary Markov chain, are presented together with the explicit expressions of the transition probabilities. They are derived by similarity transforming exactly solvable `matrix' quantum mechanics, which is recently proposed by Odake and the author. The ($q$-)Askey-scheme of hypergeometric orthogonal polynomials of a discrete variable and their dual polynomials play a central role. The most generic solvable birth/death rates are rational functions of $q^x$ ($x$ being the population) corresponding to the $q$-Racah polynomial.