This paper discusses a set of polynomials, {φ r ( s )}, orthogonal over a discrete range, with binomial distribution, b ( s ; n , p ), as the weighting function. Two recurrence relations are derived. One expresses φ r in terms of φ r -1 and Δφ r -1 , while the other relates φ r with φ r -1 and φ r -2 . It is shown that these polynomials are solutions of a finite difference equation. Also considered are two special cases. The first is the set of Hermite polynomials derived as a limiting case of the binomial-weighted orthogonal polynomials. The second deals with the Poisson distribution used as the weighting function.