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Isaac Newton Institute for Mathematical Sciences

Matrix vaued orthogonal polynomials and random walks

15th February 2007

Author: Alberto Grunbaum (University of California, Berkeley)


There is a classical result going at least as far back as Karlin and McGregor (around 1950) giving the transition probability p(x,y,t) or the N-step transition probability for a birth-and-death process in terms of some scalar valued orthogonal polynomials arising from the corresponding tridiagonal matrix. I will introduce the theory and some examples of matrix valued orthogonal polynomials, review the results of Karlin-McGregor, and show how these matrix valued polynomails allow one to deal with Markov chains that go beyond birth-and-death processes.