Moment representations of exceptional orthogonal polynomials.


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 X1-Laguerre polynomial sequence, where the constant polynomial is omitted, and the Type I X2-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 X1 sequence, and find a new flag to generate the X2 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 Xm exceptional Laguerre polynomials. We derive two representations for the X1 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 X2 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 X1 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 X1 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.



Orthogonal polynomials. Exceptional orthogonal polynomials. Moment representations. Determinantal representations. Darboux transform. Exceptional Laguerre orthogonal polynomials. Exceptional Jacobi orthogonal polynomials.