Connections between discriminants and the root distribution of polynomials with rational generating function

Abstract

Let Hm(z) be a sequence of polynomials whose generating function ∞ m=0 Hm(z)t m is the reciprocal of a bivariate polynomial D(t, z). We show that in the three cases D(t, z) = 1 + B(z)t + A(z)t 2 , D(t, z) = 1 + B(z)t + A(z)t 3 and D(t, z) = 1 + B(z)t + A(z)t 4 , where A(z) and B(z) are any polynomials in z with complex coefficients, the roots of Hm(z) lie on a portion of a real algebraic curve whose equation is explicitly given. The proofs involve the q-analogue of the discriminant, a concept introduced by Mourad Ismail.

Hyperbolic polynomials and linear-type generating functions

“…Naturally, depending on the approach, the methods used to investigate the problem can be quite different. In this paper we follow the approach employed in the works [3], [4], [9], [10] and [11] as we analyze the zero location of a sequence of polynomials {H m (z)} ∞ m=1 generated by the relation where P and Q are real stable polynomials with some restrictions on their zero locus. In two of our recent papers considering such problems the choice of the generating functions was largely motivated by the theory of multiplier sequences (and stability preserving linear operators in general).…”

The root distribution of polynomials with a three-term recurrence

“…Proof The identity (16) follows from the definition of P θ (ζ) in (3) and basic asymptotic computations. We differentiate P θ (ζ) and get…”

“…For the study of the root distribution of other sequences of polynomials that satisfy three-term recurrences, see [8,14]. In [16], the author shows that in the three special cases when n = 2, 3, and 4, the roots of H m (z) which satisfies A(z) = 0 will lie on the curve C defined in Theorem 1, and are dense there as m → ∞. This paper shows that for any fixed integer n, this result holds for all large m in the theorem below.…”

Pair correlation of roots of rational functions with rational generating functions and quadratic denominators

“…The first part of the theorem, stating that the set of roots of all H m (z) is dense on a fixed curve C has been established in [22]. For the sake of completeness, we will present a proof in Section 2.…”

“…The root distribution of certain sequences of rational functions with complex coefficients has some peculiar connections with the discriminant of the denominator of its generating function. In particular, one of the authors [22] showed that if the generating function of a sequence of polynomials is the reciprocal of a quadratic or cubic polynomial, then the roots of the corresponding sequence of polynomials form a dense set on an explicit algebraic curve. Moreover, in several cases, the endpoints of the curve are the roots of the discriminant of the denominator.…”