On Polynomial Eigenfunctions for a Class of Differential Operators

Jacobi polynomials are solutions of the differential equation (z − 1)f ′′(z) + (az + b)f ′(z) + cf(z) = 0, (1) where a, b, c are constants satisfying a > b, a + b > 0 and c = n(1 − a − n) for some nonnegative integer n. It is a classical fact that the zeros of the Jacobi polynomials lie in the interval [−1, 1], and that their density in this interval is proportional to 1/ √ 1 − |z|2 in the limit when the degree n tends to infinity. The usual proof of this statement involves the observation that, for fixed a and b, the Jacobi polynomials constitute an orthogonal system of polynomials with respect to a certain weight function on the interval [−1, 1]. The desired conclusion then follows from the general theory of orthogonal systems of polynomials. The following appears to be a natural generalization of the differential equation (1). Let k ≥ 2 be an integer, and let Q0, . . . , Qk be polynomials in one complex variable satisfying deg Qj ≤ j with equality when j = k. Moreover, we make a normalization by assuming that Qk is monic. Consider the differential operator