Abstract. Consider the Wronskians of the classical Hermite polynomialsshowed that for a special class of partitions the corresponding polynomials are orthogonal and dense among all polynomials with respect to a certain inner product, but in contrast to the usual case have some degrees missing (so-called exceptional orthogonal polynomials). We generalise their results to all partitions by considering complex contours of integration and non-positive Hermitian products. The corresponding polynomials are orthogonal and dense in a finite-codimensional subspace of C[x] satisfying certain quasiinvariance conditions. A Laurent version of exceptional orthogonal polynomials, related to monodromy-free trigonometric Schrödinger operators, is also presented. Mathematics Subject Classification. 33C47 (81Q05).

Abstract. We study Lamé operators of the formwith m ∈ N and ω a half-period of ℘(z). For rectangular period lattices, we can choose ω and z 0 such that the potential is real, periodic and regular. It is known after Ince that the spectrum of the corresponding Lamé operator has a band structure with not more than m gaps. In the first part of the paper, we prove that the opened gaps are precisely the first m ones. In the second part, we study the Lamé spectrum for a generic period lattice when the potential is complex-valued. We concentrate on the m = 1 case, when the spectrum consists of two regular analytic arcs, one of which extends to infinity, and briefly discuss the m = 2 case, paying particular attention to the rhombic lattices.