We present two infinite sequences of polynomial eigenfunctions of a Sturm-Liouville problem. As opposed to the classical orthogonal polynomial systems, these sequences start with a polynomial of degree one. We denote these polynomials as X 1 -Jacobi and X 1 -Laguerre and we prove that they are orthogonal with respect to a positive definite inner product defined over the compact interval [−1, 1] or the half-line [0, ∞), respectively, and they are a basis of the corresponding L 2 Hilbert spaces. Moreover, we prove a converse statement similar to Bochner's theorem for the classical orthogonal polynomial systems: if a self-adjoint second-order operator has a complete set of polynomial, then it must be either the X 1 -Jacobi or the X 1 -Laguerre SturmLiouville problem. A Rodrigues-type formula can be derived for both of the X 1 polynomial sequences.

We prove an extension of Bochner's classical result that characterizes the classical polynomial families as eigenfunctions of a second-order differential operator with polynomial coefficients. The extended result involves considering differential operators with rational coefficients and the requirement is that they have a numerable sequence of polynomial eigenfunctions p 1 , p 2 , . . . of all degrees except for degree zero. The main theorem of the paper provides a characterization of all such differential operators. The existence of such differential operators and polynomial sequences is based on the concept of exceptional polynomial subspaces, and the converse part of the main theorem rests on the classification of codimension one exceptional subspaces under projective transformations, which is performed in this paper.

We prove that every rational extension of the quantum harmonic oscillator that is exactly solvable by polynomials is monodromy free, and therefore can be obtained by applying a finite number of state-deleting Darboux transformations on the harmonic oscillator. Equivalently, every exceptional orthogonal polynomial system of Hermite type can be obtained by applying a Darboux-Crum transformation to the classical Hermite polynomials. Exceptional Hermite polynomial systems only exist for even codimension 2m, and they are indexed by the partitions λ of m. We provide explicit expressions for their corresponding orthogonality weights and differential operators and a separate proof of their completeness. Exceptional Hermite polynomials satisfy a 2ℓ + 3 recurrence relation where ℓ is the length of the partition λ. Explicit expressions for such recurrence relations are given.

Abstract. We adapt the notion of the Darboux transformation to the context of polynomial Sturm-Liouville problems. As an application, we characterize the recently described X m Laguerre polynomials in terms of an isospectral Darboux transformation. We also show that the shape-invariance of these new polynomial families is a direct consequence of the permutability property of the Darboux-Crum transformation.

It has been recently discovered that exceptional families of Sturm-Liouville orthogonal polynomials exist, that generalize in some sense the classical polynomials of Hermite, Laguerre and Jacobi. In this paper we show how new families of exceptional orthogonal polynomials can be constructed by means of multiple-step algebraic Darboux transformations. The construction is illustrated with an example of a 2-step Darboux transformation of the classical Laguerre polynomials, which gives rise to a new orthogonal polynomial system indexed by two integer parameters. For particular values of these parameters, the classical Laguerre and the type II $X_\ell$-Laguerre polynomials are recovered.Comment: corrected minor mistake

We investigate the backward Darboux transformations (addition of a lowest bound state) of shape-invariant potentials on the line, and classify the subclass of algebraic deformations, those for which the potential and the bound states are simple elementary functions. A countable family, m = 0, 1, 2, . . ., of deformations exists for each family of shape-invariant potentials. We prove that the m th deformation is exactly solvable by polynomials, meaning that it leaves invariant an infinite flag of polynomial modules P (m) m ⊂ P (m) m+1 ⊂ . . ., where P (m) nis a codimension m subspace of 1, z, . . . , z n . In particular, we prove that the first (m = 1) algebraic deformation of the shape-invariant class is precisely the class of operators preserving the infinite flag of exceptional monomial modules P (1) n = 1, z 2 , . . . , z n . By construction, these algebraically deformed Hamiltonians do not have an sl(2) hidden symmetry algebra structure.

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a b s t r a c tIn this paper we state and prove some properties of the zeros of exceptional Jacobi and Laguerre polynomials. Generically, the zeros of exceptional polynomials fall into two classes: the regular zeros, which lie in the interval of orthogonality and the exceptional zeros, which lie outside that interval. We show that the regular zeros have two interlacing properties: one is the natural interlacing between zeros of consecutive polynomials as a consequence of their Sturm-Liouville character, while the other one shows interlacing between the zeros of exceptional and classical polynomials. A Heine-Mehler type formula is provided for the exceptional polynomials, which allows to derive the asymptotic behaviour of their regular zeros for large degree n and fixed codimension m. We also describe the location and the asymptotic behaviour of the m exceptional zeros, which converge for large n to fixed values.