We investigate the zeros of a family of hypergeometric polynomials Mn (x; β, c) = (β)n 2F1 (-n, -x; β; 1 – 1/c), n ϵ 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 ϵ N and c = p/p-1, the polynomials Kn (x;p, N) = (- N)n 2 F1 (-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 ϵ R are real-rooted when c 0.
The family of general Jacobi polynomials P(α,β)
n where α, β ∈ C can be characterised by complex (non-
Hermitian) orthogonality relations (cf. Kuijlaars et al. (2005)). The special subclass of Jacobi polynomials
Jordaan, Kerstin Heidrun; Wang, H.; Zhou, J.(Taylor and Francis, 2014-09)
The monotonicity properties of all the zeros with respect to a parameter of orthogonal polynomials
associated with an even weight function are studied. The results we obtain extend the
work of A. Markoff. The monotonicity ...
In this paper, we prove the quasi-orthogonality of a family of 2F2 polynomials and several classes of 3F2 polynomials that do not appear in the Askey scheme for hypergeometric orthogonal polynomials. Our results include, ...