Bounds for zeros of Meixner and Kravchuk polynomials

Abstract

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.

Recurrence equations involving different orthogonal polynomial sequences and applications

“…, n − 1}, whose zeros can be used as inner bounds for the extreme zeros of p n . The bounds obtained in this way are more accurate than the inner bounds obtained using mixed recurrence equations with m = 2, as was done for the extreme zeros of the Jacobi, Laguerre and Gegenbauer polynomials in [17], Meixner and Krawtchouk polynomials in [23] and Hahn polynomials in [24]. In our applications the polynomials g n−m,k , m ∈ {2, 3, .…”

“…Remark 5. In [23] the authors use equations like (8), with m = 2, to obtain inner bounds for the Meixner polynomials. In the proof of [23,Theorem 2.1], the assumption is made that if q is any polynomial, such that b a q(x)p n (x)w(x)dx = 0, n ∈ {3, 4, .…”

Krawtchouk Polynomials and Quadratic Semi-Regular Sequences

“…Recall that we are interested in the smallest integer k ≤ 2m − n that satisfies (12). Hence, under the non-negativity condition (13) we have…”

Inner bounds for the extreme zeros of 3F2 hypergeometric polynomials

“…In order to find more precise inner bounds for the extreme zeros of a polynomial in an orthogonal sequence, the study of completed Stieltjes interlacing of zeros of different orthogonal sequences, where the different sequences are obtained by integer shifts of the parameters of the appropriate polynomials, can be helpful. This was done for the Gegenbauer, Laguerre and Jacobi polynomials in [4], the Meixner and Krawtchouk polynomials in [5] and the Pseudo-Jacobi polynomials in [6]. Mixed three-term recurrence relations satisfied by the polynomials under consideration and obtained from the connection between the appropriate polynomials, their hypergeometric representations, as well as contiguous function relations satisfied by these polynomials, are used to obtain these bounds and a Maple computer package [7] for computing contiguous relations of exclusively 2 F 1 series is helpful in this regard.…”