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The Adelic Grassmannian and Exceptional Hermite Polynomials

Abstract

It is shown that when dependence on the second flow of the KP hierarchy is added, the resulting semi-stationary wave function of certain points in George Wilson’s adelic Grassmannian are generating functions of the exceptional Hermite orthogonal polynomials. This surprising correspondence between different mathematical objects that were not previously known to be so closely related is interesting in its own right, but also proves useful in two ways: it leads to new algorithms for effectively computing the associated differential and difference operators and it also answers some open questions about them.

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References

  1. 1.

    Bakalov, B., Horozov, E., Yakimov, M.: General methods for constructing bispectral operators. Phys. Lett. A 222, 59–66 (1996)

  2. 2.

    Bochner, S.: Über Sturm-Liouvillesche Polynomsysteme. Math. Zeitschrift 29, 730–736 (1929)

  3. 3.

    Bonneux, N., Dunning, C., Stevens, M.: Coefficients of Wronskian Hermite polynomials. Studies in Applied Mathematics (2019)

  4. 4.

    Duistermaat, J.J., Grünbaum, F.A.: Differential Equations in the Spectral Parameter. Commun. Math. Phys. 103, 177–240 (1986)

  5. 5.

    Durán, A.J.: Higher order recurrence relation for exceptional Charlier, Meixner, Hermite and Laguerre orthogonal polynomials. Integr. Transf. Spec. Funct., 26, 357–376 (2015)

  6. 6.

    Durán, A.J.: Exceptional Charlier and Hermite polynomials. J. Approx. Theory 182, 29–58 (2014)

  7. 7.

    Gómez-Ullate, D., Grandati, Y, Milson, R: Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials. J. Phys. A 47(1), 015203 (2013)

  8. 8.

    Gómez-Ullate, D., Grandati, Y., Milson, R.: Durfee rectangles and pseudo-Wronskian equivalences for Hermite polynomials. Stud. Appl. Math. 141, 596–625 (2018)

  9. 9.

    García-Ferrero, M.A., Gómez-Ullate, D., Milson, R.: A Bochner type characterization theorem for exceptional orthogonal polynomials. J. Math. Anal. Appl. 472, 584–626 (2019)

  10. 10.

    Gómez-Ullate, D., Kasman, A., Kuijlaars, A.B., Milson, R.: Recurrence relations for exceptional Hermite polynomials. J. Approx. Theory 204, 1–16 (2016)

  11. 11.

    Gómez-Ullate, D., Grandati, Y., McIntyre, Z., Milson, R.: Ladder operators and rational extensions. arXiv:1910.12648 (2019)

  12. 12.

    Gómez-Ullate, D., Kamran, N., Milson, R.: An extended class of orthogonal polynomials defined by a Sturm-Liouville problem. J. Math. Anal. Appl. 359(1), 352–367 (2009)

  13. 13.

    Grünbaum, F.A., Haine, L.: Orthogonal polynomials satisfying differential equations: the role of the Darboux transformation. CRM Proceedings and Lecture Notes. Vol. 9. AMS (1996)

  14. 14.

    Grünbaum, F.A., Yakimov, M.: Discrete bispectral Darboux transformations from Jacobi operators. Pacific J. Math. 204(2), 395–431 (2002)

  15. 15.

    Haine, L., Iliev, P.: Commutative rings of difference operators and an adelic flag manifold. Int. Math. Res. Not. 2000.6, 281–s323 (2000)

  16. 16.

    Iliev, P.: Bispectral extensions of the Askey-Wilson polynomials. J. Funct. Anal. 266, 2294–2318 (2014)

  17. 17.

    Kasman, A., Rothstein, M.: Bispectral Darboux Transformations: the Generalized Airy Case. Physica D 102(3-4), 159–176 (1997)

  18. 18.

    IG, M.: Symmetric Functions and Hall Polynomials. Oxford University Press (1998)

  19. 19.

    Noumi, M.: Painlevé Equations Through Symmetry., Vol. 223. Springer Science & Business (2004)

  20. 20.

    Oblomkov, A.A.: Monodromy-free Schrödinger operators with quadratically increasing potentials. Theor. Math. Phys. 121(3), 374–386 (1999)

  21. 21.

    Odake, S., Sasaki, R.: Infinitely many shape invariant potentials and new orthogonal polynomials. Phys. Lett. B 679(4), 414–417 (2009)

  22. 22.

    Odake, S.: Recurrence relations of the multi-indexed orthogonal polynomials. J. Math. Phys. 54(8), 083506 (2013)

  23. 23.

    Rota, G.-C., Kahaner, D., Odlyzko, A.: On the foundations of combinatorial theory. VIII. Finite operator calculus. J. Math. Anal. Appl. 42.3, 684–760 (1973)

  24. 24.

    Sato, M., Sato, Y.: in Nonlinear partial differential equations in applied science (Tokyo, 1982), pp. 259–271. Amsterdam, North-Holland (1983)

  25. 25.

    Segal, G., Wilson, G.: Loop Groups and Equations of KdV Type. Publications Mathematiques, vol. 61. de l’lnstitut des Hautes Etudes Scientifiques, pp. 5–65 (1985)

  26. 26.

    Wilson, G.: Bispectral commutative ordinary differential operators. J. reine angew. Math 442, 177–204 (1993)

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Correspondence to Alex Kasman.

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Kasman, A., Milson, R. The Adelic Grassmannian and Exceptional Hermite Polynomials. Math Phys Anal Geom 23, 40 (2020). https://doi.org/10.1007/s11040-020-09365-z

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Keywords

  • Bispectrality
  • KP hierarchy
  • Exceptional orthogonal polynomials
  • Generating function
  • Hermite polynomials
  • Adelic grassmannian

Mathematics Subject Classification (2010)

  • 33C45
  • 33C47