On zeros of quasi-orthogonal Meixner polynomials

For each fixed value of β in the range −2 < β < −1 and 0 < c < 1, we investigate interlacing properties of the zeros of polynomials of consecutive degree for Mn(x;β,c) and Mk(x,β + t,c), k ∈ {n−1,n,n+1} and t ∈ {0,1,2}. We prove the conjecture in [9] on a lower bound for the first positive zero of the quasi-orthogonalorder1polynomial Mn(x;β+1,c)andidentifyupperandlowerboundsforthefirstfew zeros of quasi-orthogonal order 2 Meixner polynomials Mn(x;β,c). We show that a sequence of Meixner polynomials {Mn(x;β,c)}∞n=3 with −2 < β < −1 and 0 < c < 1 cannot be orthogonal with respect to any positive measure by proving that the zeros of Mn−1(x;β,c) and Mn(x;β,c) do not interlace for any n∈N  3.