The electrostatic equilibrium problem for classical discrete orthogonal polynomials

Rutka, Przemysław; Smarzewski, Ryszard

doi:10.1090/mcom/3808

The electrostatic equilibrium problem for classical discrete orthogonal polynomials
HTML articles powered by AMS MathViewer

by Przemysław Rutka and Ryszard Smarzewski;

Math. Comp. 92 (2023), 1331-1348

DOI: https://doi.org/10.1090/mcom/3808

Published electronically: January 27, 2023

HTML | PDF | Request permission

Abstract:

We present two types of generic difference characterizations of zeros of the classical discrete orthogonal polynomials associated with a discrete weight function satisfying a difference equation of the Pearson type. These characterizations use either the total forward or modified partial central difference operators of discrete discriminants, which approximate the total electrostatic energy of the system of $n$ unit charges moving freely on an interval in presence of external potential given by the iterated discrete weight.

References

W. A. Al-Salam and T. S. Chihara, Another characterization of the classical orthogonal polynomials, SIAM J. Math. Anal. 3 (1972), 65–70. MR 316772, DOI 10.1137/0503007
Iván Area, Dimitar K. Dimitrov, Eduardo Godoy, and Vanessa G. Paschoa, Zeros of classical orthogonal polynomials of a discrete variable, Math. Comp. 82 (2013), no. 282, 1069–1095. MR 3008850, DOI 10.1090/S0025-5718-2012-02646-9
Patrick Billingsley, Probability and measure, 2nd ed., Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. MR 830424
T. S. Chihara, An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York-London-Paris, 1978. MR 481884
David Cox, John Little, and Donal O’Shea, Using algebraic geometry, Graduate Texts in Mathematics, vol. 185, Springer-Verlag, New York, 1998. MR 1639811, DOI 10.1007/978-1-4757-6911-1
Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vols. I, II, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. MR 58756
A. G. García, F. Marcellán, and L. Salto, A distributional study of discrete classical orthogonal polynomials, Proceedings of the Fourth International Symposium on Orthogonal Polynomials and their Applications (Evian-Les-Bains, 1992), 1995, pp. 147–162. MR 1340932, DOI 10.1016/0377-0427(93)E0241-D
Á. P. Horváth, The electrostatic properties of zeros of exceptional Laguerre and Jacobi polynomials and stable interpolation, J. Approx. Theory 194 (2015), 87–107. MR 3325523, DOI 10.1016/j.jat.2015.02.004
Mourad E. H. Ismail, An electrostatics model for zeros of general orthogonal polynomials, Pacific J. Math. 193 (2000), no. 2, 355–369. MR 1755821, DOI 10.2140/pjm.2000.193.355
Mourad E. H. Ismail, Difference equations and quantized discriminants for $q$-orthogonal polynomials, Adv. in Appl. Math. 30 (2003), no. 3, 562–589. MR 1973957, DOI 10.1016/S0196-8858(02)00547-X
Mourad E. H. Ismail, Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, vol. 98, Cambridge University Press, Cambridge, 2005. With two chapters by Walter Van Assche; With a foreword by Richard A. Askey. MR 2191786, DOI 10.1017/CBO9781107325982
Mourad E. H. Ismail, Inna Nikolova, and Plamen Simeonov, Difference equations and discriminants for discrete orthogonal polynomials, Ramanujan J. 8 (2004), no. 4, 475–502 (2005). MR 2130522, DOI 10.1007/s11139-005-0276-z
Charles Jordan, Calculus of finite differences, 3rd ed., Chelsea Publishing Co., New York, 1965. Introduction by Harry C. Carver. MR 183987
Samuel Karlin and William J. Studden, Tchebycheff systems: With applications in analysis and statistics, Pure and Applied Mathematics, Vol. XV, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1966. MR 204922
Alexander Kobel and Michael Sagraloff, On the complexity of computing with planar algebraic curves, J. Complexity 31 (2015), no. 2, 206–236. MR 3305993, DOI 10.1016/j.jco.2014.08.002
Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw, Hypergeometric orthogonal polynomials and their $q$-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010. With a foreword by Tom H. Koornwinder. MR 2656096, DOI 10.1007/978-3-642-05014-5
Wolfram Koepf and Mohammad Masjed-Jamei, A generic polynomial solution for the differential equation of hypergeometric type and six sequences of orthogonal polynomials related to it, Integral Transforms Spec. Funct. 17 (2006), no. 8, 559–576. MR 2246501, DOI 10.1080/10652460600725234
Wolfram Koepf and Dieter Schmersau, Representations of orthogonal polynomials, J. Comput. Appl. Math. 90 (1998), no. 1, 57–94. MR 1627168, DOI 10.1016/S0377-0427(98)00023-5
Wolfram Koepf and Dieter Schmersau, Recurrence equations and their classical orthogonal polynomial solutions, Appl. Math. Comput. 128 (2002), no. 2-3, 303–327. Orthogonal systems and applications. MR 1891025, DOI 10.1016/S0096-3003(01)00078-9
F. Marcellán, A. Martínez-Finkelshtein, and P. Martínez-González, Electrostatic models for zeros of polynomials: old, new, and some open problems, J. Comput. Appl. Math. 207 (2007), no. 2, 258–272. MR 2345246, DOI 10.1016/j.cam.2006.10.020
Giuseppe Mastroianni and Gradimir V. Milovanović, Interpolation processes, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2008. Basic theory and applications. MR 2457729, DOI 10.1007/978-3-540-68349-0
Arnold F. Nikiforov and Vasilii B. Uvarov, Special functions of mathematical physics, Birkhäuser Verlag, Basel, 1988. A unified introduction with applications; Translated from the Russian and with a preface by Ralph P. Boas; With a foreword by A. A. Samarskiĭ. MR 922041, DOI 10.1007/978-1-4757-1595-8
Przemysław Rutka and Ryszard Smarzewski, Complete solution of the electrostatic equilibrium problem for classical weights, Appl. Math. Comput. 218 (2012), no. 10, 6027–6037. MR 2873079, DOI 10.1016/j.amc.2011.11.084
Przemysław Rutka and Ryszard Smarzewski, Explicit barycentric formulae for osculatory interpolation at roots of classical orthogonal polynomials, Math. Comp. 86 (2017), no. 307, 2409–2427. MR 3647964, DOI 10.1090/mcom/3184
Przemysław Rutka and Ryszard Smarzewski, Difference inequalities and barycentric identities for classical discrete iterated weights, Math. Comp. 88 (2019), no. 318, 1791–1804. MR 3925485, DOI 10.1090/mcom/3396
T. J. Stieltjes, Sur les polynômes de Jacobi, C. R. Hebd. Séances Acad. Sci. 100 (1885), 620–622.
T. J. Stieltjes, Sur quelques théorèmes d’Algèbre, C. R. Hebd. Séances Acad. Sci. 100 (1885), 439–440.
Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, RI, 1975. MR 372517
Galliano Valent and Walter Van Assche, The impact of Stieltjes’ work on continued fractions and orthogonal polynomials: additional material, Proceedings of the International Conference on Orthogonality, Moment Problems and Continued Fractions (Delft, 1994), 1995, pp. 419–447. MR 1379147, DOI 10.1016/0377-0427(95)00128-X
W. Van Assche, The impact of Stieltjes work on continued fractions and orthogonal polynomials, Collected Papers (G. van Dijk, ed.), vol. I, Springer, Berlin, 1993, pp. 5–37.
Haiyong Wang, Daan Huybrechs, and Stefan Vandewalle, Explicit barycentric weights for polynomial interpolation in the roots or extrema of classical orthogonal polynomials, Math. Comp. 83 (2014), no. 290, 2893–2914. MR 3246814, DOI 10.1090/S0025-5718-2014-02821-4
Haiyong Wang and Shuhuang Xiang, On the convergence rates of Legendre approximation, Math. Comp. 81 (2012), no. 278, 861–877. MR 2869040, DOI 10.1090/S0025-5718-2011-02549-4