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 β n > 0, δ n + 1 > 0, n = 0, 1, …, but δ 0 ≦ 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 { Q n (x) } generated by