1. Introduction

Consider a birth and death process, i.e., a discrete time Markov chain on the nonnegative integers, with a one step transition probability matrix ℙ. There is then a time-honored way of writing down the n-step transition probability matrix ℙn in terms of the orthogonal polynomials associated to ℙ and the spectral measure. This goes back to Karlin and McGregor [1957] and, as they observe, it is nothing but an application of the spectral theorem. One can find some precursors of these powerful ideas, see for instance [Harris 1952; Ledermann and Reuter 1954]. Inasmuch as this is such a deep and general result, it holds in many setups, such as a nearest neighbours random walk on the Nth-roots of unity. In general this representation of ℙn allows one to relate properties of the Markov chain, such as recurrence or other limiting behaviour, to properties of the orthogonality measure.

In the few cases when one can get one's hands on the orthogonality measure and the polynomials themselves this gives fairly explicit answers to various questions.

The two main drawbacks to the applicability of this representation (to be recalled below) are:

• a) typically one cannot get either the polynomials or the measure explicitly.

• b) the method is restricted to “nearest neighbour” transition probability chains that give rise to tridiagonal matrices and thus to orthogonal polynomials.