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

We classify the differential operators $T=z^j{\mathrm{\partial }}_z^j+z^m{\mathrm{\partial }}_z^{\mathrm{\ell }}$, where $0\le m<\mathrm{\ell }$ and $1\le j<\mathrm{\ell }$, whose monic eigenpolynomials satisfy a finite-term recurrence relation. Writing $k=\mathrm{\ell }-m$, such a recurrence exists if and only if $j=1$ and $k\mid \mathrm{\ell }$. In that case we determine all recurrence coefficients in closed form and prove that the associated difference operator has order $\mathrm{\ell }$. 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.