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Orthogonal Polynomials: Current Trends and Applications pp 171–192Cite as

Stable Equilibria for the Roots of the Symmetric Continuous Hahn and Wilson Polynomials

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Part of the book series: SEMA SIMAI Springer Series ((SEMA SIMAI,volume 22))

Abstract

We show that the gradient flows associated with a recently found family of Morse functions converge exponentially to the roots of the symmetric continuous Hahn polynomials. By symmetry reduction the rate of the exponential convergence can be improved, which is clarified by comparing with corresponding gradient flows for the roots of the Wilson polynomials.

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References

  1. Askey, R., Wilson, J.: A set of hypergeometric orthogonal polynomials. SIAM J. Math. Anal. 13, 651–655 (1982)

    Article  MathSciNet  Google Scholar 

  2. Beltrán, C., Marcellán, F., Martínez-Finkelshtein, A.: Some extremal properties of the roots of orthogonal polynomials. Gac. R. Soc. Mat. Esp. 21, 345–366 (2018)

    MathSciNet  MATH  Google Scholar 

  3. Bihun, O., Calogero, F.: Properties of the zeros of the polynomials belonging to the Askey scheme. Lett. Math. Phys. 104, 1571–1588 (2014)

    Article  MathSciNet  Google Scholar 

  4. Bihun, O., Calogero, F.: Properties of the zeros of the polynomials belonging to the q-Askey scheme. J. Math. Anal. Appl. 433, 525–542 (2016)

    Article  MathSciNet  Google Scholar 

  5. Calogero, F.: Equilibrium configuration of the one-dimensional n-body problem with quadratic and inversely quadratic pair potentials. Lett. Nuovo Cimento (2) 20, 251–253 (1977)

    Google Scholar 

  6. Chicone, C.: Ordinary Differential Equations with Applications. 2nd edn. Springer, New York (2006)

    MATH  Google Scholar 

  7. Dimitrov, D.K., Van Assche, W.: Lamé differential equations and electrostatics. Proc. Amer. Math. Soc. 128, 3621–3628 (2000)

    Article  MathSciNet  Google Scholar 

  8. Fehér, L., Görbe, T.F.: Duality between the trigonometric BC n Sutherland system and a completed rational Ruijsenaars-Schneider-van Diejen system. J. Math. Phys. 55(10), 102704 (2014)

    Article  MathSciNet  Google Scholar 

  9. Forrester, P.J., Rogers, J.B.: Electrostatics and the zeros of the classical polynomials. SIAM J. Math. Anal. 17, 461–468 (1986)

    Article  MathSciNet  Google Scholar 

  10. Grünbaum, F.A.: Variations on a theme of Heine and Stieltjes: an electrostatic interpretation of the zeros of certain polynomials. J. Comput. Appl. Math. 99, 189–194 (1998)

    Article  MathSciNet  Google Scholar 

  11. Grünbaum, F.A.: Electrostatic interpretation for the zeros of certain polynomials and the Darboux process. J. Comput. Appl. Math. 133, 397–412 (2001)

    Article  MathSciNet  Google Scholar 

  12. Hendriksen, E., van Rossum, H., Electrostatic interpretation of zeros. In: Orthogonal Polynomials and their Applications. Lecture Notes in Mathematics, vol. 1329, pp. 241–250. Springer, Berlin (1988)

    Google Scholar 

  13. Horváth, Á.P.: The electrostatic properties of zeros of exceptional Laguerre and Jacobi polynomials and stable interpolation. J. Approx. Theory 194, 87–107 (2015)

    Article  MathSciNet  Google Scholar 

  14. Ismail, M.E.H.: An electrostatics model for zeros of general orthogonal polynomials. Pacific J. Math. 193, 355–369 (2000)

    Article  MathSciNet  Google Scholar 

  15. Ismail, M.E.H.: More on electrostatic models for zeros of orthogonal polynomials. Numer. Funct. Anal. Optim. 21, 191–204 (2000)

    Article  MathSciNet  Google Scholar 

  16. Ismail, M.E.H.: Classical and Quantum Orthogonal Polynomials in One Variable. Cambridge University Press, Cambridge (2005)

    Book  Google Scholar 

  17. Jooste, A., Njionou Sadjang, P., Koepf, W.: Inner bounds for the extreme zeros of 3 F 2 hypergeometric polynomials. Integral Transforms Spec. Funct. 28, 361–373 (2017)

    Article  MathSciNet  Google Scholar 

  18. Khalil, H.K.: Nonlinear Systems, 3rd edn. Prentice Hall, Upper Saddle River (2002)

    Google Scholar 

  19. Koekoek, R., Lesky, P.A., Swarttouw, R.: Hypergeometric Orthogonal Polynomials and their q-Analogues. Springer, Berlin (2010)

    Book  Google Scholar 

  20. Koornwinder, T.H.: Quadratic transformations for orthogonal polynomials in one and two variables. In: Representation Theory, Special Functions and Painlevé Equations–RIMS 2015. Advanced Studies in Pure Mathematics, vol. 76, pp. 419–447. Mathematical Society of Japan, Tokyo (2018)

    Google Scholar 

  21. Marcellán, F., Martínez-Finkelshtein, A., Martínez-González, P.: Electrostatic models for zeros of polynomials: old, new, and some open problems. J. Comput. Appl. Math. 207, 258–272 (2007)

    Article  MathSciNet  Google Scholar 

  22. Odake, S., Sasaki, R.: Equilibria of ‘discrete’ integrable systems and deformation of classical orthogonal polynomials. J. Phys. A 37, 11841–11876 (2004)

    Article  MathSciNet  Google Scholar 

  23. Odake, S., Sasaki, R.: Equilibrium positions, shape invariance and Askey-Wilson polynomials. J. Math. Phys. 46(6), 063513 (2005)

    Article  MathSciNet  Google Scholar 

  24. Odake, S., Sasaki, R.: Calogero-Sutherland-Moser systems, Ruijsenaars-Schneider-van Diejen systems and orthogonal polynomials. Prog. Theor. Phys. 114, 1245–1260 (2005)

    MATH  Google Scholar 

  25. Perelomov, A.M.: Equilibrium configurations and small oscillations of some dynamical systems. Ann. Inst. H. Poincaré Sect. A (N.S.) 28, 407–415 (1978)

    Google Scholar 

  26. Pusztai, B.G.: The hyperbolic BC n Sutherland and the rational BC n Ruijsenaars-Schneider-van Diejen models: Lax matrices and duality. Nuclear Phys. B 856, 528–551 (2012)

    Article  MathSciNet  Google Scholar 

  27. Pusztai, B.G.: Scattering theory of the hyperbolic BC n Sutherland and the rational BC n Ruijsenaars-Schneider-van Diejen models. Nuclear Phys. B 874, 647–662 (2013)

    Article  MathSciNet  Google Scholar 

  28. Simanek, B.: An electrostatic interpretation of the zeros of paraorthogonal polynomials on the unit circle. SIAM J. Math. Anal. 48, 2250–2268 (2016)

    Article  MathSciNet  Google Scholar 

  29. Steinerberger, S.: Electrostatic interpretation of zeros of orthogonal polynomials. Proc. Amer. Math. Soc. 146, 5323–5331 (2018)

    Article  MathSciNet  Google Scholar 

  30. Stieltjes, T.J.: Sur certains polynômes qui vérifient une équation différentielle linéaire du second ordre et sur la theorie des fonctions de Lamé. Acta Math. 6, 321–326 (1885)

    Article  MathSciNet  Google Scholar 

  31. Szegö, G.: Orthogonal Polynomials, 4th edn. American Mathematical Society, Providence (1975)

    MATH  Google Scholar 

  32. van Diejen, J.F.: Deformations of Calogero-Moser systems and finite Toda chains. Theor. Math. Phys. 99, 549–554 (1994)

    Article  MathSciNet  Google Scholar 

  33. van Diejen, J.F.: Difference Calogero-Moser systems and finite Toda chains. J. Math. Phys. 36, 1299–1323 (1995)

    Article  MathSciNet  Google Scholar 

  34. van Diejen, J.F.: Multivariable continuous Hahn and Wilson polynomials related to integrable difference systems. J. Phys. A 28, L369–L374 (1995)

    Article  MathSciNet  Google Scholar 

  35. van Diejen, J.F.: On the equilibrium configuration of the BC-type Ruijsenaars-Schneider system. J. Nonlinear Math. Phys. 12(suppl. 1), 689–696 (2005)

    Article  MathSciNet  Google Scholar 

  36. van Diejen, J.F.: Gradient system for the roots of the Askey-Wilson polynomial. Proc. Amer. Math. Soc. 147, 5239–5249 (2019)

    Article  MathSciNet  Google Scholar 

  37. van Diejen, J.F., Emsiz, E.: Solutions of convex Bethe Ansatz equations and the zeros of (basic) hypergeometric orthogonal polynomials. Lett. Math. Phys. 109, 89–112 (2019)

    Article  MathSciNet  Google Scholar 

  38. Wilson, J.A.: Some hypergeometric orthogonal polynomials. SIAM J. Math. Anal. 11, 690–701 (1980)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

It is a pleasure to thank Alexei Zhedanov for emphasizing that the Morse functions from [37], which minimize at the roots of the continuous Hahn, Wilson and Askey-Wilson polynomials, should be viewed as natural analogs of Stieltjes’ electrostatic potentials for the roots of the classical orthogonal polynomials. Thanks are also due to an anonymous referee for suggesting some important improvements in the presentation.

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Authors and Affiliations

  1. Instituto de Matemática y Física, Universidad de Talca, Talca, Chile

    Jan Felipe van Diejen

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  1. Jan Felipe van Diejen
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Correspondence to Jan Felipe van Diejen .

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Editors and Affiliations

  1. Departamento de Matemáticas, Universidad Carlos III de Madrid, Leganés, Spain

    Francisco Marcellán

  2. Departamento de Física y Matemáticas, Universidad de Alcalá, Alcalá de Henares, Madrid, Spain

    Edmundo J. Huertas

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Diejen, J.F.v. (2021). Stable Equilibria for the Roots of the Symmetric Continuous Hahn and Wilson Polynomials. In: Marcellán, F., Huertas, E.J. (eds) Orthogonal Polynomials: Current Trends and Applications. SEMA SIMAI Springer Series, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-030-56190-1_6

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