  We initiate the study of a natural generalisation of the classical
Bochner-Krall problem asking which linear ordinary differential operators
possess sequences of eigenpolynomials satisfying linear recurrence relations of
finite length; the classical case corresponds to the 3-term recurrence
relations with real coefficients subject to some extra restrictions. We
formulate a general conjecture and prove it in the first non-trivial case of
operators of order 3.
