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

We study the second-order differential operators ${\mathcal{D}}_{\mathrm{\Xi }}$ and ${\mathcal{D}}_{\mathrm{\Lambda }}$ associated with the rescaled polynomial families $({\tilde{\mathrm{\Xi }}}_n)$ and $({\tilde{\mathrm{\Lambda }}}_n)$, and more generally the polynomial sequences generated by iterating these operators from an arbitrary linear initial datum $cx-d$. We establish structural properties of ${\mathcal{D}}_{\mathrm{\Xi }}$ and ${\mathcal{D}}_{\mathrm{\Lambda }}$, 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 $({\tilde{\mathrm{\Xi }}}_n)$ and $({\tilde{\mathrm{\Lambda }}}_n)$, prove strict interlacing of consecutive zeros under explicit conditions on $d\mathrm{/}c$, 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.