Moment representations of exceptional orthogonal polynomials.

Abstract

Exceptional orthogonal polynomials (XOPs) can be viewed as an extension of their classical orthogonal polynomial counterparts. They exclude polynomials of a certain order(s) from being eigenfunctions for their corresponding exceptional differential operator, while surprisingly still forming a complete set of eigenpolynomials for that operator. In Chapters Two and Three, we examine the (so-called) Type I $X_1$-Laguerre polynomial sequence, where the constant polynomial is omitted, and the Type I $X_2$-Laguerre polynomial sequence, where both the constant and linear polynomials are omitted. In the literature, one method presented for obtaining XOPs has been to apply Gram--Schmidt orthogonalization to a so-called ``flag". For our purposes, a flag is an infinite set of mutually linearly independent vectors which span the space of a particular XOP. Here, we use a known flag to generate the $X_1$ sequence, and find a new flag to generate the $X_2$ sequence. The particular canonical flag we pick keeps both the determinantal representation and the moment recursion manageable. We also develop a flag that could be extended to the general $X_m$ exceptional Laguerre polynomials. We derive two representations for the $X_1$ polynomials in terms of moments by using determinants. The first representation in terms of the canonical moments is rather cumbersome. We introduce adjusted moments and find a second, more elegant formula. We deduce a recursion formula for the moments and the adjusted ones. The adjusted moments are also expressed via a generating function. These representations are then extended to the $X_2$ case. We explore various complicating factors that arise in moving on to XOPs of a higher co-dimension. Finally, by employing the Darboux transform, these representations in terms of adjusted moments are then generalized to any of the $X_1$ exceptional orthogonal polynomials, regardless of the underlying family (Jacobi or Laguerre). We include a recursion formula for the adjusted moments, and provide the initial adjusted moments for each system. Throughout we relate to the various examples of $X_1$ exceptional orthogonal polynomials, but we especially focus on and provide complete proofs for the Jacobi and the Type III Laguerre cases, as they are less prevalent in literature.