Askey–Wilson polynomials, quadratic harnesses and martingales

2008/12/31 by Włodek Bryc, Jacek Wesołowski
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Applied mathematics #Askey–Wilson polynomials #Bayesian Methods and Mixture Models #Classical orthogonal polynomials #Discrete mathematics #Discrete orthogonal polynomials #Gegenbauer polynomials #Hahn polynomials #Kravchuk polynomials #Martingale (probability theory) #Mathematics #Orthogonal polynomials #Orthogonality #Pure mathematics #Quadratic equation #Random Matrices and Applications #Wilson polynomials #math.CA #math.PR

paper · pdf · doi:10.1214/09-aop503

published as Annals of Probability 2010, Vol. 38, No. 3, 1221-1262 · Published in at http://dx.doi.org/10.1214/09-AOP503 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

openalex publication_date 2010/05/01 · arxiv created 2011/01/07 · arxiv updated 2014/07/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We use orthogonality measures of Askey–Wilson polynomials to construct Markov processes with linear regressions and quadratic conditional variances. Askey–Wilson polynomials are orthogonal martingale polynomials for these processes.

Citations

• The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
• Rejoinder: Gibbs Sampling, Exponential Families and Orthogonal Polynomials
• BI-POISSON PROCESS
• Free martingale polynomials
• Asymmetric simple exclusion process with open boundaries and Askey–Wilson polynomials
• Classical and Quantum Orthogonal Polynomials in One Variable