Finite-term recurrences in a generalized Bochner--Krall family

We classify the differential operators \(T=z^j\partial_z^j+z^m\partial_z^\ell\), where \(0\le m<\ell\) and \(1\le j<\ell\), whose monic eigenpolynomials satisfy a finite-term recurrence relation. Writing \(k=\ell-m\), such a recurrence exists if and only if \(j=1\) and \(k\mid\ell\). In that case we determine all recurrence coefficients in closed form and prove that the associated difference operator has order \(\ell\). We also give an explicit factorization showing that every admissible operator is of Type~(2) in Conjecture~1.10 of Horozov--Shapiro--Tater; the equality of the differential and difference orders is the conclusion predicted by their Conjecture~1.9.