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

SIGMA 14 (2018), 048, 29 pages      arXiv:1801.07980      https://doi.org/10.3842/SIGMA.2018.048
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications (OPSFA14)

### Recurrence Relations for Wronskian Hermite Polynomials

Niels Bonneux and Marco Stevens
Department of Mathematics, University of Leuven, Celestijnenlaan 200B box 2400, 3001 Leuven, Belgium

Received January 25, 2018, in final form May 09, 2018; Published online May 16, 2018

Abstract
We consider polynomials that are defined as Wronskians of certain sets of Hermite polynomials. Our main result is a recurrence relation for these polynomials in terms of those of one or two degrees smaller, which generalizes the well-known three term recurrence relation for Hermite polynomials. The polynomials are defined using partitions of natural numbers, and the coefficients in the recurrence relation can be expressed in terms of the number of standard Young tableaux of these partitions. Using the recurrence relation, we provide another recurrence relation and show that the average of the considered polynomials with respect to the Plancherel measure is very simple. Furthermore, we show that some existing results in the literature are easy corollaries of the recurrence relation.

Key words: Wronskian; Hermite polynomials; partitions; recurrence relation.

pdf (500 kb)   tex (32 kb)

References

1. Abramowitz M., Stegun I.A., Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, Vol. 55, U.S. Government Printing Office, Washington, D.C., 1972.
2. Adin R., Roichman Y., Standard Young tableaux, in Handbook of Enumerative Combinatorics, Discrete Math. Appl. (Boca Raton), CRC Press, Boca Raton, FL, 2015, 895-974, arXiv:1408.4497.
3. Andrews G.E., The theory of partitions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1998.
4. Baik J., Deift P., Suidan T., Combinatorics and random matrix theory, Graduate Studies in Mathematics, Vol. 172, Amer. Math. Soc., Providence, RI, 2016.
5. Bonneux N., Kuijlaars A.B.J., Exceptional Laguerre polynomials, Stud. Appl. Math., to appear, arXiv:1708.03106.
6. Borodin A., Olshanski G., Representations of the infinite symmetric group, Cambridge Studies in Advanced Mathematics, Vol. 160, Cambridge University Press, Cambridge, 2017.
7. Clarkson P.A., The fourth Painlevé equation and associated special polynomials, J. Math. Phys. 44 (2003), 5350-5374.
8. Clarkson P.A., Painlevé equations - nonlinear special functions, in Orthogonal Polynomials and Special Functions, Lecture Notes in Math., Vol. 1883, Springer, Berlin, 2006, 331-411.
9. Clarkson P.A., Special polynomials associated with rational solutions of the Painlevé equations and applications to soliton equations, Comput. Methods Funct. Theory 6 (2006), 329-401.
10. Curbera G.P., Durán A.J., Invariant properties for Wronskian type determinants of classical and classical discrete orthogonal polynomials under an involution of sets of positive integers, arXiv:1612.07530.
11. Durán A.J., Exceptional Charlier and Hermite orthogonal polynomials, J. Approx. Theory 182 (2014), 29-58, arXiv:1309.1175.
12. Durán A.J., Exceptional Meixner and Laguerre orthogonal polynomials, J. Approx. Theory 184 (2014), 176-208, arXiv:1310.4658.
13. 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.
14. Durán A.J., Exceptional Hahn and Jacobi orthogonal polynomials, J. Approx. Theory 214 (2017), 9-48, arXiv:1510.02579.
15. 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.
16. 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.
17. 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.
18. Gómez-Ullate D., Grandati Y., Milson R., Durfee rectangles and pseudo-Wronskian equivalences for Hermite polynomials, arXiv:1612.05514.
19. 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.
20. 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.
21. Haese-Hill W.A., Hallnäs M.A., Veselov A.P., Complex exceptional orthogonal polynomials and quasi-invariance, Lett. Math. Phys. 106 (2016), 583-606, arXiv:1509.07008.
22. Kuijlaars A.B.J., Milson R., Zeros of exceptional Hermite polynomials, J. Approx. Theory 200 (2015), 28-39, arXiv:1412.6364.
23. Miki H., Tsujimoto S., A new recurrence formula for generic exceptional orthogonal polynomials, J. Math. Phys. 56 (2015), 033502, 13 pages, arXiv:1410.0183.
24. Noumi M., Yamada Y., Symmetries in the fourth Painlevé equation and Okamoto polynomials, Nagoya Math. J. 153 (1999), 53-86, q-alg/9708018.
25. Odake S., Sasaki R., Infinitely many shape invariant potentials and new orthogonal polynomials, Phys. Lett. B 679 (2009), 414-417, arXiv:0906.0142.
26. O'Donnell R., Analysis of Boolean functions, Cambridge University Press, New York, 2014.
27. Okamoto K., Studies on the Painlevé equations. III. Second and fourth Painlevé equations, $P_{{\rm II}}$ and $P_{{\rm IV}}$, Math. Ann. 275 (1986), 221-255.
28. Stanley R.P., Differential posets, J. Amer. Math. Soc. 1 (1988), 919-961.
29. Szegő G., Orthogonal polynomials, American Mathematical Society, Colloquium Publications, Vol. 23, 4th ed., Amer. Math. Soc., Providence, R.I., 1975.
30. Van Assche W., Orthogonal polynomials and Painlevé equations, Australian Mathematical Society Lecture Series, Vol. 27, Cambridge University Press, Cambridge, 2018.
31. Young A., On quantitative substitutional analysis, Proc. Lond. Math. Soc. 33 (1901), 97-146.
32. Young A., On quantitative substitutional analysis, Proc. Lond. Math. Soc. 28 (1928), 255-292.