Zeros of Meixner and Krawtchouk polynomials

Abstract

We investigate the zeros of a family of hypergeometric polynomials 2 F 1 (−n, −x; a; t), n ∈ N that are known as the Meixner polynomials for certain values of the parameters a and t. When a = −N, N ∈ N and t = 1 p , the polynomials. . N, 0 < p < 1 are referred to as Krawtchouk polynomials. We prove results for the zero location of the orthogonal polynomials K n (x; p, a), 0 < p < 1 and a > n − 1, the quasiorthogonal polynomials K n (x; p, a), k − 1 < a < k, k = 1, . . . , n − 1 and p > 1 or p < 0 as well as the non-orthogonal polynomials K n (x; p, N), 0 < p < 1 and n = N + 1, N + 2, . . . . We also show that the polynomials K n (x; p, a), a ∈ R are real-rooted when p → 0.