On zeros of quasi-orthogonal Meixner polynomials

For each fixed value of beta in the range 2 < beta < -1 and 0 < c < 1, we investigate interlacing properties of the zeros of polynomials of consecutive degree for M-n ( x; beta, c) and M-k (x, beta broken vertical bar t, c), k subset of [n 1, n, n broken vertical bar 1] and t is an element of{0,1,2}. We prove the conjecture in [9] on a lower bound for the first positive zero of the quasi-orthogonal order 1 polynomial M-n( x; beta + 1, c) and identify upper and lower bounds for the first few zeros of quasi-orthogonal order 2 Meixner polynomials M-n( x; beta, c). We show that a sequence of Meixner polynomials f M-n (x; beta, c)(infinity) (n=3) with 2 < beta< 1 and 0 < c < 1 cannot be orthogonal with respect to any positive measure by proving that the zeros of Mn- 1( x;beta, c) and M-n (x; beta, c) do not interlace for any n is an element of N >= 3.