Structure and Zero Asymptotics of Differential Operators Associated with $Ξ_n$ and $Λ_n$

Abstract

We study the second-order differential operators \(\mathcal D_Ξ\) and \(\mathcal D_Λ\) associated with the rescaled polynomial families \((\widetildeΞ_n)\) and \((\widetildeΛ_n)\), and more generally the polynomial sequences generated by iterating these operators from an arbitrary linear initial datum \(cx-d\). We establish structural properties of \(\mathcal D_Ξ\) and \(\mathcal D_Λ\), 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 \((\widetildeΞ_n)\) and \((\widetildeΛ_n)\), 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.