Complementary Romanovski-Routh polynomials: From orthogonal polynomials on the unit circle to Coulomb wave functions
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- by A. Martínez-Finkelshtein, L. L. Silva Ribeiro, A. Sri Ranga and M. Tyaglov PDF
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Abstract:
We consider properties and applications of a sequence of polynomials known as complementary Romanovski-Routh polynomials (CRR polynomials for short). These polynomials, which follow from the Romanovski-Routh polynomials or complexified Jacobi polynomials, are known to be useful objects in the studies of the one-dimensional Schrödinger equation and also the wave functions of quarks. One of the main results of this paper is to show how the CRR-polynomials are related to a special class of orthogonal polynomials on the unit circle. As another main result, we have established their connection to a class of functions which are related to a subfamily of Whittaker functions that includes those associated with the Bessel functions and the regular Coulomb wave functions. An electrostatic interpretation for the zeros of CRR-polynomials is also considered.References
- Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1992. Reprint of the 1972 edition. MR 1225604
- George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958, DOI 10.1017/CBO9781107325937
- P. Appell, Sur une classe de polynômes, Ann. Sci. École Norm. Sup. (2) 9 (1880), 119–144 (French). MR 1508688
- Gábor Szegő, Collected papers. Vol. 1, Contemporary Mathematicians, Birkhäuser, Boston, Mass., 1982. 1915–1927; Edited by Richard Askey; Including commentaries and reviews by George Pólya, P. C. Rosenbloom, Askey, L. E. Payne, T. Kailath and Barry M. McCoy. MR 674482
- Árpád Baricz, Turán type inequalities for regular Coulomb wave functions, J. Math. Anal. Appl. 430 (2015), no. 1, 166–180. MR 3347207, DOI 10.1016/j.jmaa.2015.04.082
- C. F. Bracciali, A. Martínez-Finkelshtein, A. Sri Ranga, and D. O. Veronese, Christoffel formula for kernel polynomials on the unit circle, J. Approx. Theory 235 (2018), 46–73. MR 3845610, DOI 10.1016/j.jat.2018.05.001
- C. F. Bracciali, A. Sri Ranga, and A. Swaminathan, Para-orthogonal polynomials on the unit circle satisfying three term recurrence formulas, Appl. Numer. Math. 109 (2016), 19–40. MR 3541940, DOI 10.1016/j.apnum.2016.05.008
- K. Castillo, M. S. Costa, A. Sri Ranga, and D. O. Veronese, A Favard type theorem for orthogonal polynomials on the unit circle from a three term recurrence formula, J. Approx. Theory 184 (2014), 146–162. MR 3218796, DOI 10.1016/j.jat.2014.05.007
- 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 0481884
- M. S. Costa, H. M. Felix, and A. Sri Ranga, Orthogonal polynomials on the unit circle and chain sequences, J. Approx. Theory 173 (2013), 14–32. MR 3073604, DOI 10.1016/j.jat.2013.04.009
- Alfredo Deaño, Javier Segura, and Nico M. Temme, Computational properties of three-term recurrence relations for Kummer functions, J. Comput. Appl. Math. 233 (2010), no. 6, 1505–1510. MR 2559338, DOI 10.1016/j.cam.2008.03.051
- Dimitar K. Dimitrov and A. Sri Ranga, Zeros of a family of hypergeometric para-orthogonal polynomials on the unit circle, Math. Nachr. 286 (2013), no. 17-18, 1778–1791. MR 3145170, DOI 10.1002/mana.201200181
- Dimitar K. Dimitrov and Walter Van Assche, Lamé differential equations and electrostatics, Proc. Amer. Math. Soc. 128 (2000), no. 12, 3621–3628. MR 1709747, DOI 10.1090/S0002-9939-00-05638-0
- Carl-Erik Fröberg, Numerical treatment of Coulomb wave functions, Rev. Mod. Phys. 27 (1955), 399–411. MR 0073289, DOI 10.1103/revmodphys.27.399
- Walter Gautschi, Computational aspects of three-term recurrence relations, SIAM Rev. 9 (1967), 24–82. MR 213062, DOI 10.1137/1009002
- Yasuhiko Ikebe, The zeros of regular Coulomb wave functions and of their derivatives, Math. Comp. 29 (1975), 878–887. MR 378361, DOI 10.1090/S0025-5718-1975-0378361-5
- 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 and David R. Masson, Generalized orthogonality and continued fractions, J. Approx. Theory 83 (1995), no. 1, 1–40. MR 1354960, DOI 10.1006/jath.1995.1106
- M. E. H. Ismail and A. Sri Ranga, $R_{II}$ type recurrence, generalized eigenvalue problem and orthogonal polynomials on the unit circle, Linear Algebra Appl. 562 (2019), 63–90. MR 3886340, DOI 10.1016/j.laa.2018.10.005
- 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
- A. Martínez-Finkelshtein, A. Sri Ranga, and D. O. Veronese, Extreme zeros in a sequence of para-orthogonal polynomials and bounds for the support of the measure, Math. Comp. 87 (2018), no. 309, 261–288. MR 3716196, DOI 10.1090/mcom/3210
- A. S. Meligy, Simple expansion for the regular Coulomb wave function, Nuclear Phys. 6 (1958), 440–442.
- N. Michel, Precise Coulomb wave functions for a wide range of complex $ell$, $\eta$ and $z$, Comput. Phys. Commun. 176 (2007), 232–249.
- Yoshinori Miyazaki, Yasushi Kikuchi, DongSheng Cai, and Yasuhiko Ikebe, Error analysis for the computation of zeros of regular Coulomb wave function and its first derivative, Math. Comp. 70 (2001), no. 235, 1195–1204. MR 1710636, DOI 10.1090/S0025-5718-00-01241-2
- G. Natanson, Routh polynomials: hundred years in obscurity, preprint, 2018, https://www.researchgate.net/publication/326522529_Routh_polynomials_hundred_years_in_obscurity.
- 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
- John L. Powell, Recurrence formulas for Coulomb wave functions, Phys. Rev. (2) 72 (1947), 626–627. MR 22277
- A. P. Raposo, H. J. Weber, D. E. Alvarez-Castillo, and M. Kirchbach, Romanovski polynomials in selected physics problems, Cent. Eur. J. Phys. 5 (2007), 253–284.
- V. Romanovski, Sur quelques classes nouvelles de polynomes orthogonaux, C. R. Acad. Sci. Paris 188 (1929), 1023–1025.
- E. J. Routh, On some Properties of certain Solutions of a Differential Equation of the Second Order, Proc. Lond. Math. Soc. 16 (1884/85), 245–261. MR 1575769, DOI 10.1112/plms/s1-16.1.245
- Barry Simon, Orthogonal polynomials on the unit circle. Part 1, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Providence, RI, 2005. Classical theory. MR 2105088, DOI 10.1090/coll054.1
- Barry Simon, Orthogonal polynomials on the unit circle. Part 2, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Providence, RI, 2005. Spectral theory. MR 2105089, DOI 10.1090/coll/054.2/01
- J. R. Shepanski and S. T. Butler, An expansion for Coulomb wave functions, Nuclear Phys. 1 (1956), 313–321. MR 0093355
- L. J. Slater, Confluent hypergeometric functions, Cambridge University Press, New York, 1960. MR 0107026
- A. Sri Ranga, Szegő polynomials from hypergeometric functions, Proc. Amer. Math. Soc. 138 (2010), no. 12, 4259–4270. MR 2680052, DOI 10.1090/S0002-9939-2010-10592-0
- F. Štampach and P. Šťovíček, Orthogonal polynomials associated with Coulomb wave functions, J. Math. Anal. Appl. 419 (2014), no. 1, 231–254. MR 3217146, DOI 10.1016/j.jmaa.2014.04.049
- Stefan Steinerberger, Electrostatic interpretation of zeros of orthogonal polynomials, Proc. Amer. Math. Soc. 146 (2018), no. 12, 5323–5331. MR 3866871, DOI 10.1090/proc/14226
- Gabor Szegö, Orthogonal polynomials, American Mathematical Society Colloquium Publications, Vol. 23, American Mathematical Society, Providence, R.I., 1959. Revised ed. MR 0106295
- G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR 0010746
- Hans J. Weber, Connections between real polynomial solutions of hypergeometric-type differential equations with Rodrigues formula, Cent. Eur. J. Math. 5 (2007), no. 2, 415–427. MR 2301942, DOI 10.2478/s11533-007-0004-6
- Hans J. Weber, Connections between Romanovski and other polynomials, Cent. Eur. J. Math. 5 (2007), no. 3, 581–595. MR 2322830, DOI 10.2478/s11533-007-0014-4
- Jet Wimp, Some explicit Padé approximants for the function $\Phi ’/\Phi$ and a related quadrature formula involving Bessel functions, SIAM J. Math. Anal. 16 (1985), no. 4, 887–895. MR 793930, DOI 10.1137/0516067
- N. S. Witte and P. J. Forrester, Gap probabilities in the finite and scaled Cauchy random matrix ensembles, Nonlinearity 13 (2000), no. 6, 1965–1986. MR 1794841, DOI 10.1088/0951-7715/13/6/305
- F. L. Yost, J. A. Wheeler, and G. Breit, Coulomb wave functions in repulsive fields, Phys. Rev. 49 (1936), 174–189.
- Alexei Zhedanov, Biorthogonal rational functions and the generalized eigenvalue problem, J. Approx. Theory 101 (1999), no. 2, 303–329. MR 1726460, DOI 10.1006/jath.1999.3339
Additional Information
- A. Martínez-Finkelshtein
- Affiliation: Department of Mathematics, Baylor University, 76798 Waco, Texas; and Departamento de Matemáticas, Universidad de Almería, 04120 Alamería, Spain
- MR Author ID: 248069
- ORCID: 0000-0001-9421-5624
- Email: a_martinez-finkelshtein@baylor.edu
- L. L. Silva Ribeiro
- Affiliation: Pós-Graduação em Matemática, UNESP-Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP, Brazil
- Email: luana.math@hotmail.com
- A. Sri Ranga
- Affiliation: Departamento de Matemática Aplicada, IBILCE, UNESP-Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP, Brazil
- MR Author ID: 238837
- Email: sri.ranga@unesp.br
- M. Tyaglov
- Affiliation: School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, People’s Republic of China
- MR Author ID: 823374
- Email: tyaglov@sjtu.edu.cn
- Received by editor(s): June 7, 2018
- Received by editor(s) in revised form: June 12, 2018, and October 1, 2018
- Published electronically: March 7, 2019
- Additional Notes: The first author was partially supported by the Spanish government together with the European Regional Development Fund (ERDF) under grant MTM2017-89941-P (from MINECO), by Junta de Andalucía (the research group FQM-229), and by Campus de Excelencia Internacional del Mar (CEIMAR) of the University of Almería.
This work was started while the first and the third authors were visiting the fourth author at Shanghai Jiao Tong University in the fall of 2016. Subsequently, this work was developed as part of the Ph.D. thesis of the second author, partially supported by the grant 2017/04358-8 from FAPESP of Brazil.
The recent collaboration of the third author in this work was also supported by the grants 2016/09906-0 and 2017/12324-6 of FAPES Brazil and 305073/2014-1 of CNPq Brazil. The third author is the corresponding author
The fourth author’s work was supported by the Joint NSFC-ISF Research Program, jointly funded by the National Natural Science Foundation of China and the Israel Science Foundation (No. 11561141001). - Communicated by: Mourad Ismail
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 2625-2640
- MSC (2010): Primary 42C05, 33C45; Secondary 33C47
- DOI: https://doi.org/10.1090/proc/14423
- MathSciNet review: 3951438