Markov Chains Generated by Convolutions of Orthogonality Measures

Abstract

About two dozens of exactly solvable Markov chains on one-dimensional finite and semi-infinite integer lattices are constructed in terms of convolutions of orthogonality measures of the Krawtchouk, Hahn, Meixner, Charlier, $q$-Hahn, $q$-Meixner and little $q$-Jacobi polynomials. By construction, the stationary probability distributions, the complete sets of eigenvalues and eigenvectors are provided by the polynomials and the orthogonality measures. An interesting property possessed by these stationary probability distributions, called `convolutional self-similarity,' is demonstrated.

arXiv comment46 pages. Comments added, some repeated results deleted. To appear in J.Phys.A

References in corpus (7)

• Orthogonal Polynomials from Hermitian Matrices
• Exactly Solvable Birth and Death Processes
• Orthogonal Polynomials from Hermitian Matrices II
• A System of Multivariable Krawtchouk Polynomials and a Probabilistic Application
• Big q-Laguerre and q-Meixner polynomials and representations of the algebra U_q(su(1,1))
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Cited by in corpus (4)

• Type II Multi-indexed Little $q$-Jacobi and Little $q$-Laguerre Polynomials
• 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
• Rahman polynomials