Hyperbolic polynomials and linear-type generating functions

The zero locus and some combinatorial properties of certain exponential Sheffer sequences

“…The main result of the paper concerning the zeros of certain exponential Sheffer sequences is the following theorem. We point out that this result is stronger than the results found in a string of recent works on the zero distribution of various polynomial sequences (see for example [2], [4], [5], [6]) in that the present paper provides the exact curve on which the zeros of the P m s lie for all m, not just for m 1. We are able to establish the main result for all P m due to a simple differential recurrence relation the P m s must satisfy (see the opening discussion of Section 2), essentially identifying the the shift operator P m ∆ −→ P m−1 as scaled differentiation -a hyperbolicity preserving linear operator.…”

“…We are able to establish the main result for all P m due to a simple differential recurrence relation the P m s must satisfy (see the opening discussion of Section 2), essentially identifying the the shift operator P m ∆ −→ P m−1 as scaled differentiation -a hyperbolicity preserving linear operator. We remark that the current problem shares a heuristic trait with those studied in [4], [5], [6]. As noted in these works, the choice for the particular families of generating functions was motivated by the central role hyperbolic polynomials (and in particular the polynomials (1 + x) n ) play in the theory of hyperbolicity preserving linear operators on R[x].…”

On combinatorial properties and the zero distribution of certain Sheffer sequences

“…On one hand, the requirement that a polynomial sequence be a Sheffer sequence restricts the type of generating function the sequence may have. On the other hand, the sequences under consideration in the current paper have generating functions that are quite different from those that have been studied in some recent works (see [5], [6], [7], [10], [18], and [19]). Broadly speaking, these works study the zero distribution of sequences ps n q ně0 generated by certain 'rational'-type bivariate generating functions.…”

Generalizing Tran's Conjecture

“…One can check that in this specific case, the latter curve C given by ( 5) is exactly the Beraha-Kahane-Weiss curve Γ Q . In [6] Conjecture A was proven for k = 2, 3, 4 and in [7] Conjecture A was proven for arbitrary k, but only for polynomials P n (z) with sufficiently large n. Several other aspects of this problem are discussed in [3], [8], [9]. The purpose of this short note is to generalize and settle the first part of Conjecture A.…”