On Polynomial Eigenfunctions of a Hypergeometric-Type Operator

Consider an operator dQ(f) = d dxk (Q(x)f(x)) where Q(x) is some fixed polynomial of degree k. One can easily see that dQ has exactly one polynomial eigenfunction pn(x) in each degree n ≥ 0 and its eigenvalue λn,k equals (n+k)! n! . A more intriguing fact is that all zeros of pn(x) lie in the convex hull of the set of zeros to Q(x). In particular, if Q(x) has only real zeros then each pn(x) enjoys the same property. We formulate a number of conjectures on different properties of pn(x) based on computer experiments as, for example, the interlacing property, a formula for the asymptotic distribution of zeros etc. These polynomial egenfunctions might be thought as an interesting generalization of the classical Gegenbauer polynomials with the integer value of the parameter (which corresponds to the case Q(x) = (x2 − 1)). §