Exceptional orthogonal polynomials

With David Gomez-Ullate (ICMAT and Universidad Complutense de Madrid)

Exceptional orthogonal polynomials

Exceptional orthgonal polynomials are families of complete orthogonal polynomials that arise as eigenfunctions of a Sturm-Liouville problem, which differ from the classical families of Hermite, Laguerre and Jacobi in that there is a finite number of gaps in their degree sequence. Despite the “missing” degrees, the remaining polynomials still span a complete basis of a weighted L2 space, and the orthogonality weight is a rational modification of a classical weight.
We will briefly review the main results in the theory of exceptional orthogonal polynomials (classification, position of their zeros, recurrence relations, etc.), with emphasis on the similarities and differences with classical polynomials.
We will discuss some applications in mathematical physics and Painlevé equations and end up with some questions on their potential use for numerical analysis and approximation theory.

[1] D. Gomez-Ullate, N. Kamran, R. Milson, An extended class of orthogonal polynomials defined by a Sturm–Liouville problem, J. Math. Anal. Appl. 359 (1), 352-367 (2009).
[2] D. Gomez-Ullate, A. Kasman, A. Kuijlaars, R. Milson, Recurrence relations for exceptional Hermite polynomials, J. Approx. Theory 204, 1-16 (2016)
[3] M. A. Garcia-Ferrero, D. Gomez-Ullate, Oscillation theorems for the Wronskian of an arbitrary sequence of eigenfunctions of Schrödinger’s equation, Lett. Math. Phys. 105 (4), 551-573 (2015)
[4] D. Gomez-Ullate, Y. Grandati, R. Milson, Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials, J. Phys. A 47 (1), 015203 (2014).
[5] D. Gomez-Ullate, N. Kamran, R. Milson, A conjecture on exceptional orthogonal polynomials, Found. Comput. Math. 13 (4), 615-666 (2013)

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