We investigate the zeros of a family of hypergeometric polynomials

We consider the interlacing of zeros of polynomials within the sequences of quasi-orthogonal order one Meixner polynomials {Mn(x; β, c)} ∞ n=1 characterised by −β, c ∈ (0, 1). The interlacing of zeros of quasiorthogonal Meixner polynomials Mn(x; β, c) with the zeros of their nearest orthogonal counterparts M l (x; β + k, c), l, n ∈ N, k ∈ {1, 2}, is also discussed.

In this paper, we prove the quasi-orthogonality of a family of 2 F 2 polynomials and several classes of 3 F 2 polynomials that do not appear in the Askey scheme for hypergeometric orthogonal polynomials. Our results include, as a special case, two 3 F 2 polynomials considered by Dickinson in 1961. We also discuss the location and interlacing of the real zeros of our polynomials.

The zeros of certain different sequences of orthogonal polynomials interlace in a well-defined way. The study of this phenomenon and the conditions under which it holds lead to a set of points that can be applied as bounds for the extreme zeros of the polynomials. We consider different sequences of the discrete orthogonal Meixner and Kravchuk polynomials and use mixed threeterm recurrence relations, satisfied by the polynomials under consideration, to identify bounds for the extreme zeros of Meixner and Kravchuk polynomials.

We show how to obtain linear combinations of polynomials in an orthogonal sequence {P n } n≥0 , such as Q n,k (x) = k i=0 a n,i P n−i (x), a n,0 a n,k = 0, that characterize quasiorthogonal polynomials of order k ≤ n − 1. The polynomials in the sequence {Q n,k } n≥0 are obtained from P n , by making use of parameter shifts. We use an algorithmic approach to find these linear combinations for each family applicable and these equations are used to prove quasi-orthogonality of order k. We also determine the location of the extreme zeros of the quasi-orthogonal polynomials with respect to the end points of the interval of orthogonality of the sequence {P n } n≥0 , where possible.