Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 016, 19 pages      arXiv:2008.02822

Exceptional Legendre Polynomials and Confluent Darboux Transformations

María Ángeles García-Ferrero a, David Gómez-Ullate bc and Robert Milson d
a) Institut für Angewandte Mathematik, Ruprecht-Karls-Universität Heidelberg, Im Neunheimer Feld 205, 69120 Heidelberg, Germany
b) Departamento de Ingeniería Informática, Escuela Superior de Ingeniería, Universidad de Cádiz, 11519 Puerto Real, Spain
c) Departamento de Física Teórica, Universidad Complutense de Madrid, 28040 Madrid, Spain
d) Department of Mathematics and Statistics, Dalhousie University, Halifax, NS, B3H 3J5, Canada

Received September 22, 2020, in final form February 03, 2021; Published online February 20, 2021

Exceptional orthogonal polynomials are families of orthogonal polynomials that arise as solutions of Sturm-Liouville eigenvalue problems. They generalize the classical families of Hermite, Laguerre, and Jacobi polynomials by allowing for polynomial sequences that miss a finite number of ''exceptional'' degrees. In this paper we introduce a new construction of multi-parameter exceptional Legendre polynomials by considering the isospectral deformation of the classical Legendre operator. Using confluent Darboux transformations and a technique from inverse scattering theory, we obtain a fully explicit description of the operators and polynomials in question. The main novelty of the paper is the novel construction that allows for exceptional polynomial families with an arbitrary number of real parameters.

Key words: exceptional orthogonal polynomials; Darboux transformations; isospectral deformations.

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  1. Abraham P.B., Moses H.E., Changes in potentials due to changes in the point spectrum: anharmonic oscillators with exact solutions, Phys. Rev. A 22 (1980), 1333-1340.
  2. Akritas A.G., Akritas E.K., Malaschonok G.I., Various proofs of Sylvester's (determinant) identity, Math. Comput. Simulation 42 (1996), 585-593.
  3. Bonneux N., Exceptional Jacobi polynomials, J. Approx. Theory 239 (2019), 72-112, arXiv:1804.01323.
  4. Contreras-Astorga A., Schulze-Halberg A., Recursive representation of Wronskians in confluent supersymmetric quantum mechanics, J. Phys. A: Math. Theor. 50 (2017), 105301, 15 pages, arXiv:1702.00843.
  5. Durán A.J., Higher order recurrence relation for exceptional Charlier, Meixner, Hermite and Laguerre orthogonal polynomials, Integral Transforms Spec. Funct. 26 (2015), 357-376, arXiv:1409.4697.
  6. Durán A.J., Exceptional Hahn and Jacobi orthogonal polynomials, J. Approx. Theory 214 (2017), 9-48, arXiv:1510.02579.
  7. García-Ferrero M.A., Gómez-Ullate D., Milson R., A Bochner type characterization theorem for exceptional orthogonal polynomials, J. Math. Anal. Appl. 472 (2019), 584-626, arXiv:1603.04358.
  8. García-Ferrero M.A., Gómez-Ullate D., Milson R., Confluent Darboux transformations and exceptional orthogonal polynomials, in preparation.
  9. Gesztesy F., Teschl G., On the double commutation method, Proc. Amer. Math. Soc. 124 (1996), 1831-1840.
  10. Gómez-Ullate D., Grandati Y., Milson R., Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials, J. Phys. A: Math. Theor. 47 (2014), 015203, 27 pages, arXiv:1306.5143.
  11. Gómez-Ullate D., Kamran N., Milson R., An extended class of orthogonal polynomials defined by a Sturm-Liouville problem, J. Math. Anal. Appl. 359 (2009), 352-367, arXiv:0807.3939.
  12. Gómez-Ullate D., Kamran N., Milson R., A conjecture on exceptional orthogonal polynomials, Found. Comput. Math. 13 (2013), 615-666, arXiv:1203.6857.
  13. Gómez-Ullate D., Kasman A., Kuijlaars A.B.J., Milson R., Recurrence relations for exceptional Hermite polynomials, J. Approx. Theory 204 (2016), 1-16, arXiv:1506.03651.
  14. Gómez-Ullate D., Marcellán F., Milson R., Asymptotic and interlacing properties of zeros of exceptional Jacobi and Laguerre polynomials, J. Math. Anal. Appl. 399 (2013), 480-495, arXiv:1204.2282.
  15. Grandati Y., Bérard A., Comments on the generalized SUSY QM partnership for Darboux-Pöschl-Teller potential and exceptional Jacobi polynomials, J. Engrg. Math. 82 (2013), 161-171.
  16. Grandati Y., Quesne C., Confluent chains of DBT: enlarged shape invariance and new orthogonal polynomials, SIGMA 11 (2015), 061, 26 pages, arXiv:1503.07747.
  17. Hemery A.D., Veselov A.P., Whittaker-Hill equation and semifinite-gap Schrödinger operators, J. Math. Phys. 51 (2010), 072108, 17 pages, arXiv:0906.1697.
  18. Horváth Á.P., The electrostatic properties of zeros of exceptional Laguerre and Jacobi polynomials and stable interpolation, J. Approx. Theory 194 (2015), 87-107, arXiv:1410.0906.
  19. Keung W.-Y., Sukhatme U.P., Wang Q.M., Imbo T.D., Families of strictly isospectral potentials, J. Phys. A: Math. Gen. 22 (1989), L987-L992.
  20. Kuijlaars A.B.J., Milson R., Zeros of exceptional Hermite polynomials, J. Approx. Theory 200 (2015), 28-39, arXiv:1412.6364.
  21. Marquette I., Quesne C., New families of superintegrable systems from Hermite and Laguerre exceptional orthogonal polynomials, J. Math. Phys. 54 (2013), 042102, 16 pages, arXiv:1211.2957.
  22. Miki H., Tsujimoto S., A new recurrence formula for generic exceptional orthogonal polynomials, J. Math. Phys. 56 (2015), 033502, 13 pages, arXiv:1410.0183.
  23. Odake S., Recurrence relations of the multi-indexed orthogonal polynomials. III, J. Math. Phys. 57 (2016), 023514, 24 pages, arXiv:1509.08213.
  24. Odake S., Sasaki R., Infinitely many shape invariant potentials and new orthogonal polynomials, Phys. Lett. B 679 (2009), 414-417, arXiv:0906.0142.
  25. Post S., Tsujimoto S., Vinet L., Families of superintegrable Hamiltonians constructed from exceptional polynomials, J. Phys. A: Math. Theor. 45 (2012), 405202, 10 pages, arXiv:1206.0480.
  26. Quesne C., Solvable rational potentials and exceptional orthogonal polynomials in supersymmetric quantum mechanics, SIGMA 5 (2009), 084, 24 pages, arXiv:0906.2331.
  27. Schulze-Halberg A., Roy B., Darboux partners of pseudoscalar Dirac potentials associated with exceptional orthogonal polynomials, Ann. Physics 349 (2014), 159-170, arXiv:1409.0999.
  28. Sparenberg J.M., Baye D., Supersymmetric transformations of real potentials on the line, J. Phys. A: Math. Gen. 28 (1995), 5079-5095.
  29. Sukumar C.V., Supersymmetric quantum mechanics and the inverse scattering method, J. Phys. A: Math. Gen. 18 (1985), 2937-2955.
  30. Szegő G., Orthogonal Polynomials, American Mathematical Society Colloquium Publications, Vol. 23, Amer. Math. Soc., New York, 1939.

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