First return probabilities of birth and death chains and associated orthogonal polynomials

Dette, Holger

doi:10.1090/S0002-9939-00-05699-9

First return probabilities of birth and death chains and associated orthogonal polynomials
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Proc. Amer. Math. Soc. 129 (2001), 1805-1815 Request permission

Abstract:

For a birth and death chain on the nonnegative integers, integral representations for first return probabilities are derived. While the integral representations for ordinary transition probabilities given by Karlin and McGregor (1959) involve a system of random walk polynomials and the corresponding measure of orthogonality, the formulas for the first return probabilities are based on the corresponding systems of associated orthogonal polynomials. Moreover, while the moments of the measure corresponding to the random walk polynomials give the ordinary return probabilities to the origin, the moments of the measure corresponding to the associated polynomials give the first return probabilities to the origin. As a by-product we obtain a new characterization in terms of canonical moments for the measure of orthogonality corresponding to the first associated orthogonal polynomials. The results are illustrated by several examples.

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