On Sieved Orthogonal Polynomials II: Random Walk Polynomials

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Abstract
A birth and death process is a stationary Markov process whose states are the nonnegative integers and the transition probabilities (1.1) satisfy (1.2) as t → 0. Here we assume βn > 0, δn + 1 > 0, n = 0, 1, …, but δ0 ≦ 0. Karlin and McGregor [10], [11], [12], showed that each birth and death process gives rise to two sets of orthogonal polynomials. The first is the set of birth and death process polynomials {Qn(x)} generated by

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