In search of a higher Bochner theorem (Q6651976)

The classical Bochner-Krall problem asks which linear ordinary differential operators possess sequences of eigenpolynomials satisfying linear recurrence relations of finite length; the classical case corresponds to 3-term recurrence relations with real coefficients subject to some extra restrictions. The authors formulate the generalized Bochner-Krall problem (GBKP).\N\NTheir main result reads: All irreducible differential operators of order 3 solving GBKP whose corresponding difference operators also have order 3 are given by the two types: \N\NType 1: \(L=a_1\partial +a_2x\partial^2+a_3x^2\partial^3+x\partial\), \(a_j\in \mathbb{C}\), \(a_3\neq 0\), \(\partial =d/dx\).\NThey generate the 2-orthogonal monic polynomials and they satisfy a 4-term recurrence relation with the standard initial conditions \(P_{-2}(x)=P_{-1}(x)=0\), \(P_0(x)=1\). \N\NType 2: \(L=a_1\partial +a_2\partial^2+a_3\partial^3+x\partial\), \(a_j\in \mathbb{C}\), \(a_3\neq 0\), generating the monic Appell polynomials. They also satisfy a 4-term recurrence relation with the standard initial condition.