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Random walks and orthogonal polynomials: some challenges. (English) Zbl 1184.33004

Pinsky, Mark (ed.) et al., Probability, geometry and integrable systems. For Henry McKean’s seventy-fifth birthday. Cambridge: Cambridge University Press (ISBN 978-0-521-89527-9/hbk). Mathematical Sciences Research Institute Publications 55, 241-260 (2008).
S. Karlin and J. McGregor [Pac. J. Math. 1, Trans. Am. Math. Soc. 86, 366–400 (1957; Zbl 0091.13802)] worked out that discrete time Markov chains on the non-negative integers are related to orthogonal polynomials with their orthogonality measure and their three-term recurrence relations. This article gives a general overview of this approach by concrete examples, and it points out various directions how this approach can be generalized. In particular, examples are presented how important Markov chains arising from physics can be handled by matrix-valued orthogonal polynomials.
For the entire collection see [Zbl 1144.35004].

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods

Citations:

Zbl 0091.13802
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