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

SIGMA 17 (2021), 003, 14 pages      arXiv:2006.15534

The Expansion of Wronskian Hermite Polynomials in the Hermite Basis

Codruţ Grosu a and Corina Grosu b
a) Google Zürich, Brandschenkestrasse 110, Zürich, Switzerland
b) Department of Applied Mathematics, Politehnica University of Bucharest, Splaiul Independentei 313, Bucharest, Romania

Received July 08, 2020, in final form January 04, 2021; Published online January 09, 2021

We express Wronskian Hermite polynomials in the Hermite basis and obtain an explicit formula for the coefficients. From this we deduce an upper bound for the modulus of the roots in the case of partitions of length 2. We also derive a general upper bound for the modulus of the real and purely imaginary roots. These bounds are very useful in the study of irreducibility of Wronskian Hermite polynomials. Additionally, we generalize some of our results to a larger class of polynomials.

Key words: Wronskian; Hermite polynomials; Schrödinger operator.

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  1. Ben Romdhane N., On the zeros of $d$-symmetric $d$-orthogonal polynomials, J. Math. Anal. Appl. 344 (2008), 888-897.
  2. Bonneux N., Asymptotic behavior of Wronskian polynomials that are factorized via $p$-cores and $p$-quotients, Math. Phys. Anal. Geom. 23 (2020), 36, 27 pages, arXiv:2005.03516.
  3. Bonneux N., Dunning C., Stevens M., Coefficients of Wronskian Hermite polynomials, Stud. Appl. Math. 144 (2020), 245-288, arXiv:1909.03874.
  4. Bonneux N., Hamaker Z., Stembridge J., Stevens M., Wronskian Appell polynomials and symmetric functions, Adv. in Appl. Math. 111 (2019), 101932, 23 pages, arXiv:1812.01864.
  5. Bonneux N., Stevens M., Recurrence relations for Wronskian Hermite polynomials, SIGMA 14 (2018), 048, 29 pages, arXiv:1801.07980.
  6. Clarkson P.A., Special polynomials associated with rational and algebraic solutions of the Painlevé equations, in Théories asymptotiques et équations de Painlevé, Sémin. Congr., Vol. 14, Soc. Math. France, Paris, 2006, 21-52.
  7. Clarkson P.A., Gómez-Ullate D., Grandati Y., Milson R., Cyclic Maya diagrams and rational solutions of higher order Painlevé systems, Stud. Appl. Math. 144 (2020), 357-385, arXiv:1811.09274.
  8. Crum M.M., Associated Sturm-Liouville systems, Quart. J. Math. Oxford Ser. (2) 6 (1955), 121-127.
  9. 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.
  10. Fulton W., Harris J., Representation theory: a first course, Graduate Texts in Mathematics, Vol. 129, Springer-Verlag, New York, 1991.
  11. García-Ferrero M.A., Gómez-Ullate D., Oscillation theorems for the Wronskian of an arbitrary sequence of eigenfunctions of Schrödinger's equation, Lett. Math. Phys. 105 (2015), 551-573, arXiv:1408.0883.
  12. Gómez-Ullate D., Grandati Y., Milson R., Durfee rectangles and pseudo-Wronskian equivalences for Hermite polynomials, Stud. Appl. Math. 141 (2018), 596-625, arXiv:1612.05514.
  13. Gómez-Ullate D., Grandati Y., Milson R., Complete classification of rational solutions of $A_{2n}$-Painlevé systems, arXiv:2010.00076.
  14. 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.
  15. 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.
  16. Grosu C., Grosu C., The irreducibility of some Wronskian Hermite polynomials, arXiv:2007.00065.
  17. Kasman A., Milson R., The adelic Grassmannian and exceptional Hermite polynomials, Math. Phys. Anal. Geom. 23 (2020), 40, 51 pages, arXiv:2006.10025.
  18. Kuijlaars A.B.J., Milson R., Zeros of exceptional Hermite polynomials, J. Approx. Theory 200 (2015), 28-39, arXiv:1412.6364.
  19. Milovanović G.V., Rassias T.M., Inequalities for polynomial zeros, in Survey on classical inequalities, Math. Appl., Vol. 517, Kluwer Acad. Publ., Dordrecht, 2000, 165-202.
  20. Noumi M., Yamada Y., Symmetries in the fourth Painlevé equation and Okamoto polynomials, Nagoya Math. J. 153 (1999), 53-86, arXiv:q-alg/9708018.
  21. Oblomkov A.A., Monodromy-free Schrödinger operators with quadratically increasing potentials, Theoret. and Math. Phys. 121 (1999), 1574-1584.
  22. Rainville E.D., Special functions, The Macmillan Company, New York, 1960.
  23. Stanley R.P., Enumerative combinatorics, Vol. 2, Cambridge Studies in Advanced Mathematics, Vol. 62, Cambridge University Press, Cambridge, 1999.
  24. Szegő G., Orthogonal polynomials, Colloquium Publications, Vol. 23, 4th ed., Amer. Math. Soc., Providence, R.I., 1975.
  25. Turán P., Hermite-expansion and strips for zeros of polynomials, Arch. Math. 5 (1954), 148-152.
  26. Walsh J.L., An inequality for the roots of an algebraic equation, Ann. of Math. 25 (1924), 285-286.

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