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

SIGMA 18 (2022), 047, 16 pages      arXiv:2202.01278      https://doi.org/10.3842/SIGMA.2022.047

Determinantal Formulas for Exceptional Orthogonal Polynomials

Brian Simanek
Baylor University Math Department, Waco, Texas, 76706, USA

Received February 11, 2022, in final form June 17, 2022; Published online June 25, 2022

Abstract
We present determinantal formulas for families of exceptional $X_m$-Laguerre and exceptional $X_m$-Jacobi polynomials and also for exceptional $X_2$-Hermite polynomials. The formulas resemble Vandermonde determinants and use the zeros of the classical orthogonal polynomials.

Key words: exceptional orthogonal polynomials; determinantal formulas.

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References

  1. Bonneux N., Exceptional Jacobi polynomials, J. Approx. Theory 239 (2019), 72-112, arXiv:1804.01323.
  2. Bonneux N., Kuijlaars A.B.J., Exceptional Laguerre polynomials, Stud. Appl. Math. 141 (2018), 547-595, arXiv:1708.03106.
  3. Durán A.J., Exceptional Charlier and Hermite orthogonal polynomials, J. Approx. Theory 182 (2014), 29-58, arXiv:1309.1175.
  4. Durán A.J., Exceptional Meixner and Laguerre orthogonal polynomials, J. Approx. Theory 184 (2014), 176-208, arXiv:1310.4658.
  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. Durán A.J., Corrigendum to the papers on Exceptional orthogonal polynomials: J. Approx. Theory 182 (2014) 29-58, 184 (2014), 176-208 and 214 (2017), 9-48, J. Approx. Theory 253 (2020), 105349, 5 pages.
  8. Durán A.J., The algebra of recurrence relations for exceptional Laguerre and Jacobi polynomials, Proc. Amer. Math. Soc. 149 (2021), 173-188.
  9. Durán A.J., Christoffel transform of classical discrete measures and invariance of determinants of classical and classical discrete polynomials, J. Math. Anal. Appl. 503 (2021), 125306, 29 pages.
  10. Durán A.J., Pérez M., Admissibility condition for exceptional Laguerre polynomials, J. Math. Anal. Appl. 424 (2015), 1042-1053, arXiv:1409.4901.
  11. Felder G., Hemery A.D., Veselov A.P., Zeros of Wronskians of Hermite polynomials and Young diagrams, Phys. D 241 (2012), 2131-2137, arXiv:1005.2695.
  12. 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.
  13. García-Ferrero M.A., Gómez-Ullate D., Milson R., Exceptional Legendre polynomials and confluent Darboux transformations, SIGMA 17 (2021), 016, 19 pages, arXiv:2008.02822.
  14. 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.
  15. Gómez-Ullate D., Grandati Y., Milson R., Corrigendum on the proof of completeness for exceptional Hermite polynomials, J. Approx. Theory 253 (2020), 105350, 10, arXiv:1911.10602.
  16. 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.
  17. 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.
  18. 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.
  19. Ho C.L., Odake S., Sasaki R., Properties of the exceptional $(X_\ell)$ Laguerre and Jacobi polynomials, SIGMA 7 (2011), 107, 24 pages, arXiv:0912.5447.
  20. Kelly J.S., Liaw C., Osborn J., Moment representations of exceptional $X_1$ orthogonal polynomials, J. Math. Anal. Appl. 455 (2017), 1848-1869, arXiv:1610.06531.
  21. Kelly J.S., Liaw C., Osborn J., Moment representations of Type I $X_2$ exceptional Laguerre polynomials, Adv. Dyn. Syst. Appl. 14 (2019), 49-65, arXiv:1705.07851.
  22. Kuijlaars A.B.J., McLaughlin K.T.R., Riemann-Hilbert analysis for Laguerre polynomials with large negative parameter, Comput. Methods Funct. Theory 1 (2001), 205-233, arXiv:math.CA/0204248.
  23. Kuijlaars A.B.J., Milson R., Zeros of exceptional Hermite polynomials, J. Approx. Theory 200 (2015), 28-39, arXiv:1412.6364.
  24. Liaw C., Littlejohn L., Kelly J.S., Spectral analysis for the exceptional $X_m$-Jacobi equation, Electron. J. Differential Equations 2015 (2015), 194, 10 pages, arXiv:1501.04698.
  25. Liaw C., Littlejohn L.L., Milson R., Stewart J., The spectral analysis of three families of exceptional Laguerre polynomials, J. Approx. Theory 202 (2016), 5-41, arXiv:1407.4145.
  26. Liaw C., Osborn J., Moment representations of the exceptional $X_1$-Laguerre orthogonal polynomials, Math. Nachr. 290 (2017), 1716-1731, arXiv:1506.07580.
  27. Olver F.W.J., Olde Daalhuis A.B., Lozier D.W., Schneider B.I., Boisvert R.F., Clark C.W., Miller B.R., Saunders B.V., Cohl H.S., McClain M.A. (Editors), NIST Digital Library of Mathematical Functions, Release 1.1.5 of 2022-03-15, https://dlmf.nist.gov/.
  28. Quesne C., Solvable rational potentials and exceptional orthogonal polynomials in supersymmetric quantum mechanics, SIGMA 5 (2009), 084, 24 pages, arXiv:0906.2331.
  29. Simanek B., Convergence rates of exceptional zeros of exceptional orthogonal polynomials, Comput. Methods Funct. Theory, to appear, arXiv:2009.09432.
  30. SzegHo G., Orthogonal polynomials, 4th ed., American Mathematical Society Colloquium Publications, Vol. 23, Amer. Math. Soc., Providence, R.I., 1975.

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