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arXiv · 1610.06531

Moment Representations of Exceptional $X_1$ Orthogonal Polynomials

Constanze Liaw·Jessica Stewart Kelly·John Osborn
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Abstract

We obtain representations of $X_1$ exceptional orthogonal polynomials through determinants of matrices that have certain adjusted moments as entries. We start out directly from the Darboux transformation, allowing for a universal perspective, rather than dependent upon the particular system (Jacobi or Type of Laguerre polynomials). 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. We especially focus on and provide complete proofs for the Jacobi and the Type III Laguerre case, as they are less prevalent in literature. Lastly, we include a preliminary discussion explaining that the higher co-dimension setting becomes more involved. The number of possibilities and choices is exemplified, but only starts, with the lack of a canonical flag.

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Constanze Liaw, Jessica Stewart Kelly, John Osborn. 2016-10-20. Moment Representations of Exceptional $X_1$ Orthogonal Polynomials. https://doi.org/10.1016/j.jmaa.2017.05.037

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