%0 Journal Article
%T Exactly Solvable Birth and Death Processes
%A Ryu Sasaki
%J Physics
%D 2009
%I arXiv
%R 10.1063/1.3215983
%X 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.
%U http://arxiv.org/abs/0903.3097v1