Structure and Zero Asymptotics of Differential Operators Associated with ${\Xi}_n$ and ${\Lambda}_n$

TL;DR

This paper analyzes second-order differential operators linked to specific polynomial families, revealing their structural properties, zero-preserving features, and asymptotic zero distribution behavior.

Contribution

It provides a detailed structural and asymptotic analysis of differential operators associated with polynomial families generated by iteration.

Findings

Operators can be factorized into first-order components.

They preserve hyperbolicity and zeros within certain intervals.

Zero counting measures converge to a common limiting distribution.

Abstract

We study the second-order differential operators \(\mathcal D_{\Xi}\) and \(\mathcal D_{\Lambda}\) associated with the rescaled polynomial families \((\widetilde{\Xi}_n)\) and \((\widetilde{\Lambda}_n)\), and more generally the polynomial sequences generated by iterating these operators from an arbitrary linear initial datum \(cx-d\). We establish structural properties of \(\mathcal D_{\Xi}\) and \(\mathcal D_{\Lambda}\), including factorizations into first-order operators, weighted divergence forms, formal self-adjointness, and hypergeometric descriptions of the corresponding formal eigenvalue equations. We also show that both operators preserve hyperbolicity, preserve zeros in \((0,b)\) for \(b\ge 1\), and preserve proper position. For the iterated polynomial sequences, we derive explicit closed formulae in terms of the auxiliary families \((\widetilde{\Xi}_n)\) and \((\widetilde{\Lambda}_n)\), prove strict interlacing of consecutive zeros under explicit conditions on \(d/c\), and obtain asymptotic formulae for the normalized logarithmic derivatives. As a consequence, the associated zero counting measures converge weakly to the same limiting probability measure as in the auxiliary case.