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
Askey, R., Wilson, J.: A set of hypergeometric orthogonal polynomials. SIAM J. Math. Anal. 13, 651–655 (1982)
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)
Bihun, O., Calogero, F.: Properties of the zeros of the polynomials belonging to the Askey scheme. Lett. Math. Phys. 104, 1571–1588 (2014)
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)
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)
Chicone, C.: Ordinary Differential Equations with Applications. 2nd edn. Springer, New York (2006)
Dimitrov, D.K., Van Assche, W.: Lamé differential equations and electrostatics. Proc. Amer. Math. Soc. 128, 3621–3628 (2000)
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)
Forrester, P.J., Rogers, J.B.: Electrostatics and the zeros of the classical polynomials. SIAM J. Math. Anal. 17, 461–468 (1986)
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)
Grünbaum, F.A.: Electrostatic interpretation for the zeros of certain polynomials and the Darboux process. J. Comput. Appl. Math. 133, 397–412 (2001)
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)
Horváth, Á.P.: The electrostatic properties of zeros of exceptional Laguerre and Jacobi polynomials and stable interpolation. J. Approx. Theory 194, 87–107 (2015)
Ismail, M.E.H.: An electrostatics model for zeros of general orthogonal polynomials. Pacific J. Math. 193, 355–369 (2000)
Ismail, M.E.H.: More on electrostatic models for zeros of orthogonal polynomials. Numer. Funct. Anal. Optim. 21, 191–204 (2000)
Ismail, M.E.H.: Classical and Quantum Orthogonal Polynomials in One Variable. Cambridge University Press, Cambridge (2005)
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)
Khalil, H.K.: Nonlinear Systems, 3rd edn. Prentice Hall, Upper Saddle River (2002)
Koekoek, R., Lesky, P.A., Swarttouw, R.: Hypergeometric Orthogonal Polynomials and their q-Analogues. Springer, Berlin (2010)
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)
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)
Odake, S., Sasaki, R.: Equilibria of ‘discrete’ integrable systems and deformation of classical orthogonal polynomials. J. Phys. A 37, 11841–11876 (2004)
Odake, S., Sasaki, R.: Equilibrium positions, shape invariance and Askey-Wilson polynomials. J. Math. Phys. 46(6), 063513 (2005)
Odake, S., Sasaki, R.: Calogero-Sutherland-Moser systems, Ruijsenaars-Schneider-van Diejen systems and orthogonal polynomials. Prog. Theor. Phys. 114, 1245–1260 (2005)
Perelomov, A.M.: Equilibrium configurations and small oscillations of some dynamical systems. Ann. Inst. H. Poincaré Sect. A (N.S.) 28, 407–415 (1978)
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)
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)
Simanek, B.: An electrostatic interpretation of the zeros of paraorthogonal polynomials on the unit circle. SIAM J. Math. Anal. 48, 2250–2268 (2016)
Steinerberger, S.: Electrostatic interpretation of zeros of orthogonal polynomials. Proc. Amer. Math. Soc. 146, 5323–5331 (2018)
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)
Szegö, G.: Orthogonal Polynomials, 4th edn. American Mathematical Society, Providence (1975)
van Diejen, J.F.: Deformations of Calogero-Moser systems and finite Toda chains. Theor. Math. Phys. 99, 549–554 (1994)
van Diejen, J.F.: Difference Calogero-Moser systems and finite Toda chains. J. Math. Phys. 36, 1299–1323 (1995)
van Diejen, J.F.: Multivariable continuous Hahn and Wilson polynomials related to integrable difference systems. J. Phys. A 28, L369–L374 (1995)
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)
van Diejen, J.F.: Gradient system for the roots of the Askey-Wilson polynomial. Proc. Amer. Math. Soc. 147, 5239–5249 (2019)
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)
Wilson, J.A.: Some hypergeometric orthogonal polynomials. SIAM J. Math. Anal. 11, 690–701 (1980)
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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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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