On the zeros of Meixner polynomials

We investigate the zeros of a family of hypergeometric polynomials Mn(x;β, c) = (β)n 2F1(−n,−x;β; 1 − 1c ), n ∈ N, known as Meixner polynomials, that are orthogonal on (0,∞) with respect to a discrete measure for β > 0 and 0 < c < 1. When β = −N , N ∈ N and c = p p−1 , the polynomials Kn(x; p,N) = (−N)n 2F1(−n,−x;−N ; 1 p ), n = 0, 1, . . . N , 0 < p < 1 are referred to as Krawtchouk polynomials. We prove results for the zero location of the orthogonal polynomials Mn(x;β, c), c < 0 and n < 1 − β, the quasiorthogonal polynomials Mn(x;β, c), −k < β < −k + 1, k = 1, . . . , n − 1 and 0 < c < 1 or c > 1, as well as the polynomials Kn(x; p,N) with non-Hermitian orthogonality for 0 < p < 1 and n = N + 1, N + 2, . . . . We also show that the polynomials Mn(x;β, c), β ∈ R are real-rooted when c→ 0.