Exactly Solvable Birth and Death Processes

Details Exactly Solvable Birth and Death Processes Exactly Solvable Birth and Death Processes

2009-03-18

arxiv.org/abs/0903.3097v1

Physics

Mathematical Physics

LaTeX, amsmath, amssymb, 24 pages, no figures

Scientific paper

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.

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