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

For a birth and death chain on the nonnegative integers, integral represent ations for first return probabilities are derived. While the integral repre sentations for ordinary transition probabilities given by Karlin and McGreg or (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 wa lk polynomials give the ordinary return probabilities to the origin, the mo ments of the measure corresponding to the associated polynomials give the f irst return probabilities to the origin. As a by-product we obtain a new characterization in terms of canonical mome nts for the measure of orthogonality corresponding to the first associated orthogonal polynomials. The results are illustrated by several examples.