Professor David Gómez-Ullate, a member of our IE Research Datalab, recently visited the University of Granada to participate in the IMAG Conference on Orthogonal Polynomials, Special Functions, and Applications (OPSFA17). The conference provided a platform for mathematicians, physicists, and computational scientists to share recent research findings in the areas of orthogonal polynomials and special functions.
During the event, Professor Gómez-Ullate gave a talk titled “Classification of Exceptional Jacobi Polynomials”. David began by presenting a conjecture regarding the connection between Darboux transformations and exceptional orthogonal polynomials (XOPs) [1], explaining how it was later confirmed in his research [2]. This work laid the foundation for the classification of XOPs but uncovered new complexities with the discovery of additional families of exceptional polynomials, each defined by a set of continuous parameters.
These new families stem from a certain degeneracy found in Laguerre and Jacobi polynomials under specific parameter values. With further study of this mechanism, a complete classification of exceptional Jacobi polynomials -including both the traditional and new families- can be achieved.
Professor Gómez-Ullate referred to his previous research [3, 4], which demonstrates that some exceptional polynomial families are continuous deformations of classical polynomials, such as Legendre and ultraspherical polynomials. The most general class of the new exceptional Jacobi families can be seen as a deformation of known exceptional Jacobi families indexed by two partitions [5]. This classification is based on linking spectral diagrams to operators, a topic further explored by Robert Milson in his presentation.
References:
[1] D. Gómez-Ullate, N. Kamran, and R. Milson, “A conjecture on exceptional orthogonal polynomials”, Foundations of Computational Mathematics 13 (2013), 615–666.
[2] M. Á. García-Ferrero, D. Gómez-Ullate, and R. Milson, “A Bochner-type characterization theorem for exceptional orthogonal polynomials”, J. Math. Anal. Appl. 472 (2019), 584–626.
[3] M. Á. García-Ferrero, D. Gómez-Ullate, and R. Milson, “Exceptional Legendre polynomials and confluent Darboux transformations”, SIGMA 17 (2021), 016.
[4] M. Á. García-Ferrero, D. Gómez-Ullate, R. Milson, and J. Munday, “Exceptional Gegenbauer polynomials via isospectral deformation”, Studies in Applied Mathematics 149(2) (2022), 324-363.
[5] N. Bonneux, “Exceptional Jacobi polynomials”, J. Approx. Theory 239 (2019), 72–112.