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Asymptotic Behavior of Wronskian Polynomials that are Factorized via p-cores and p-quotients

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Abstract

In this paper we consider Wronskian polynomials labeled by partitions that can be factorized via the combinatorial concepts of p-cores and p-quotients. We obtain the asymptotic behavior for these polynomials when the p-quotient is fixed while the size of the p-core grows to infinity. For this purpose, we associate the p-core with its characteristic vector and let all entries of this vector simultaneously tend to infinity. This result generalizes the Wronskian Hermite setting which is recovered when p = 2.

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References

  1. Ayyer, A., Sinha, S.: The size of t-cores and hook lengths of random cells in random partitions, Preprint arXiv:1911.03135 (2019)

  2. Bertola, M., Bothner, T.: Zeros of large degree Vorob’ev–Yablonski polynomials via a Hankel determinant identity. Int. Math. Res. Notices. 2015, 9330–9399 (2015)

    Article  MathSciNet  Google Scholar 

  3. Bessenrodt, C., Gramain, J., Olsson, J.B.: Generalized hook lengths in symbols and partitions. J. Algebr. Comb. 36, 309–332 (2012)

    Article  MathSciNet  Google Scholar 

  4. Bonneux, N., Dunning, C., Stevens, M.: Coefficients of Wronskian Hermite polynomials. Stud. Appl. Math. 144, 245–288 (2020)

    Article  MathSciNet  Google Scholar 

  5. Bonneux, N., Hamaker, Z., Stembridge, J., Stevens, M.: Wronskian appell polynomials and symmetric functions. Adv. Appl. Math. 111, 101932 (2019)

    Article  MathSciNet  Google Scholar 

  6. Bonneux, N., Stevens, M.: Recurrence relations for Wronskian Hermite polynomials. Sym. Integ. Geo. Meth. Appl. 14, 048, 29 (2018)

    MathSciNet  MATH  Google Scholar 

  7. Brunat, O., Nath, R.: Cores and quotients of partitions through the Frobenius symbol, Preprint arXiv:1911.12098 (2019)

  8. Buckingham R.J.: Large-degree asymptotics of rational Painlevé-IV functions associated to generalized Hermite polynomials, To appear in Int. Math. Res. Notices (2018)

  9. Buckingham, R.J., Miller, P.D.: Large-degree asymptotics of rational Painlevé,-II functions: noncritical behaviour. Nonlinearity 27, 2489–2577 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  10. Buckingham, R.J., Miller, P.D.: Large-degree asymptotics of rational Painlevé,-II functions: critical behaviour. Nonlinearity 28, 1539–1596 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  11. Clarkson, P.A.: On rational solutions of the fourth painlevé equation and its Hamiltonian. In: Gómez-Ullate, D., Iserles, A., Levi, D., Olver, P.J., Quispel, R., Tempesta, P., Winternitz, P. (eds.) Group theory and numerical analysis, CRM proceedings and lecture notes, american mathematical society, providence, Rhode Island, vol. 39, pp 103–118 (2005)

  12. 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, 357–385 (2020)

    Article  MathSciNet  Google Scholar 

  13. Date, E., Jimbo, M., Miwa, T.: Solitons: differential equations, symmetries and infinite dimensional algebras Cambridge Tracks in Mathematics, vol. 135. Cambridge University Press, Cambridge (2000)

    Google Scholar 

  14. Durán, A. J.: Higher order recurrence relation for exceptional Charlier, Meixner, Hermite and Laguerre orthogonal polynomials. Int. Trans. Special Funct. 26, 357–376 (2015)

    Article  MathSciNet  Google Scholar 

  15. Garvan, F., Kim, D., Stanton, D.: Cranks and t-cores. Inventiones Math. 101, 1–17 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  16. Gómez-Ullate, D., Grandati, Y., Milson, R.: Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials. J. Phys. A. Math. Theor. 47, 015203, 27 (2013)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  18. James, G., Kerber, A.: The representation theory of the symmetric group. Encyclopedia of mathematics and its applications, vol. 16. Addison-Wesley Publishing Company, Reading (1981)

  19. Kajiwara, K., Ohta, Y.: Determinant structure of the rational solutions for the Painlevé, II equation. J. Math. Phys. 37, 4393–4704 (1996)

    Article  Google Scholar 

  20. Kajiwara, K., Ohta, Y.: Determinant structure of the rational solutions for the Painlevé, IV equation. J. Phys. A Math. General 31, 2431–2446 (1998)

    Article  ADS  Google Scholar 

  21. Kuijlaars, A.B.J., Milson, R.: Zeros of exceptional Hermite polynomials. J. Approx. Theor. 200, 28–39 (2015)

    Article  MathSciNet  Google Scholar 

  22. Macdonald, I.G.: Symmetric functions and hall polynomials, oxford mathematical monographs, 2nd edn. Oxford University Press, New York (1995)

    Google Scholar 

  23. Masoero, D., Roffelsen, P.: Poles of Painlevé IV rationals and their distribution. Symmetry Integr. Geo. Methods Appl. 14, 002, 49 (2018)

    MATH  Google Scholar 

  24. Masoero, D., Roffelsen, P.: Roots of generalised Hermite polynomials when both parameters are large, Preprint arXiv:1907.08552 (2019)

  25. Noumi, M., Yamada, Y.: Symmetries in the fourth Painlevé, equation and Okamoto polynomials. Nagoya Math. J. 153, 53–86 (1999)

    Article  MathSciNet  Google Scholar 

  26. Nath, R.: Advances in the theory of cores and simultaneous core partitions. Amer. Math.Monthly 124, 844–861 (2017)

    Article  MathSciNet  Google Scholar 

  27. Van Assche, W.: Orthogonal polynomials and painlevé equations, vol. 27. Cambridge University Press, Cambridge (2018)

    MATH  Google Scholar 

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Acknowledgments

The author is thankful to Dan Betea, Arno Kuijlaars and Marco Stevens for their valuable feedback and carefully reading several versions of this paper. This work is supported in part by the long term structural funding – Methusalem grant of the Flemish Government, and by EOS project 30889451 of the Flemish Science Foundation (FWO).

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  1. Department of Mathematics, KU Leuven, Leuven, 3001, Belgium

    Niels Bonneux

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Bonneux, N. Asymptotic Behavior of Wronskian Polynomials that are Factorized via p-cores and p-quotients. Math Phys Anal Geom 23, 36 (2020). https://doi.org/10.1007/s11040-020-09358-y

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