On Roots of Eigenpolynomials for Degenerate Exactly-Solvable Differential Operators

Abstract

In this paper we partially settle our conjecture from [1] (math.SP/0701143) on roots of eigenpolynomials for degenerate exactly-solvable operators. Namely, for any such operator, we establish a lower bound (which supports our conjecture) for the largest modulus of the roots of its unique and monic eigenpolynomial as its degree tends to infinity. The main theorem below thus extends earlier results obtained in [1] for a more restrictive class of operators.