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 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.

arXiv comment29 pages, no figure

References in corpus (10)

• Orthogonal Polynomials from Hermitian Matrices
• Multi-indexed (q-)Racah Polynomials
• The Exceptional (X_{\ell}) (q)-Racah Polynomials
• Exactly Solvable Birth and Death Processes
• Orthogonal Polynomials from Hermitian Matrices II
• Big q-Laguerre and q-Meixner polynomials and representations of the algebra U_q(su(1,1))
• Duality of q-polynomials, orthogonal on countable sets of points
• Hahn polynomials and the Burnside process
• A spectral analysis of discrete-time quantum walks related to the birth and death chains
• A set of orthogonal polynomials, dual to alternative q-Charlier polynomials

Cited by in corpus (3)

• Markov Chains Generated by Convolutions of Orthogonality Measures
• Quantum vs Classical Birth and Death Processes; Exactly Solvable Examples
• Multi-indexed Orthogonal Polynomials of a Discrete Variable and Exactly Solvable Birth and Death Processes