x = 0, 1, * , n 1. From this it follows when a, d> -1 that, if y is an integer > mn, the zeros of Pm(` 7)(x) are real and simple and lie in the open interval (0, y 1). In the present paper this conclusion is extended to all real -y > mn and also to 7y < -(im + ae + d) with (d + -y, -a 1) as the interval containing the zeros in the latter case. If the inequalities on a, f, -y are relaxed, the conclusion fails. In fact, the zeros need no longer be simple or even real. We also show that under the same conditions consecutive zeros are spaced more than one unit apart. These results are obtained in ?5. In ?6 we prove a number of separation theorems for the zeros of Hahn polynomials of different degrees or with different parameter values. In the earlier sections we derive various formulas partly for later use and partly for their own interest. In particular, in ?3 we establish some symmetry properties and in ?4 derive an expression for the difference APm(`0 ") (x), which we use to give a simplified form to the summation formula of Bartko [13], and to construct from Hahn polynomials an Appell set of the second kind. The Hahn polynomials arose as a limiting form in the treatment of some general systems of orthogonal polynomials by Hahn [1]. Some important particular cases