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Complementary Romanovski–Routh Polynomials, Orthogonal Polynomials on the Unit Circle, and Extended Coulomb Wave Functions

Abstract

In a recent paper (Martínez-Finkelshtein et al. in Proc Am Math Soc 147:2625–2640, 2019) some interesting results were obtained concerning complementary Romanovski–Routh polynomials, a class of orthogonal polynomials on the unit circle and extended regular Coulomb wave functions. The class of orthogonal polynomials here are generalization of the class of circular Jacobi polynomials. In the present paper, in addition to looking at some further properties of the complementary Romanovski–Routh polynomials and associated orthogonal polynomials on the unit circle, behaviour of the zeros of these extended Coulomb wave functions are also studied.

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References

  1. 1.

    Abromowitz, M., Stegun, I.A. (eds): Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards, Applied Mathematics Series—55, tenth printing (1972)

  2. 2.

    Andrews, G.E., Askey, R., Roy, R.: Special functions. In: Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M. (eds.) Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2000)

  3. 3.

    Bracciali, C.F., Sri Ranga, A., Swaminathan, A.: Para-orthogonal polynomials on the unit circle satisfying three term recurrence formulas. Appl. Numer. Math. 109, 19–40 (2016)

  4. 4.

    Baricz, Á.: Turán type inequalities for regular Coulomb wave functions. J. Math. Anal. Appl. 430, 166–180 (2015)

  5. 5.

    Baricz, Á., Štampach, F.: The Hurwitz-type theorem for the regular Coulomb wave function via Hankel determinants. Linear Algebra Appl. 548, 259–272 (2018)

  6. 6.

    Bracciali, C.F., Martínez-Finkelshtein, A., Sri Ranga, A., Veronese, D.O.: Christoffel formula for kernel polynomials on the unit circle. J. Approx. Theory 235, 46–73 (2018)

  7. 7.

    Chihara, T.S.: An Introduction to Orthogonal Polynomials: Gordon and Breach, Mathematics and its Applications Series (1978)

  8. 8.

    Deaño, A., Segura, J., Temme, N.: Computational properties of three-term recurrence relations for Kummer functions. J. Comput. Appl. Math. 233, 1505–1510 (2010)

  9. 9.

    Dzieciol, A., Yngve, S., Fröman, P.O.: Coulomb wave functions with complex values of the variable and the parameters. J. Math. Phys. 40, 6145–6166 (1999)

  10. 10.

    Fröberg, C.E.: Numerical treatment of Coulomb wave functions. Rev. Mod. Phys. 27, 399–411 (1955)

  11. 11.

    Gautschi, W.: Computational aspects of three-term recurrence relations. SIAM Rev. 9, 24–82 (1967)

  12. 12.

    Humblet, J.: Analytical structure and properties of Coulomb wave functions for real and complex energies. Ann. Phys. 155, 461–493 (1984)

  13. 13.

    Ikebe, Y.: The zeros of regular Coulomb wave functions and of their derivatives. Math. Comp. 29, 878–887 (1975)

  14. 14.

    Ismail, M.E.H.: Classical and quantum orthogonal polynomials in one variable. In: Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M. (eds.) Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (2005)

  15. 15.

    Ismail, M.E.H., Masson, D.R.: Generalized orthogonality and continued fractions. J. Approx. Theory 83, 1–40 (1995)

  16. 16.

    Ismail, M.E.H., Sri Ranga, A.: \(R_{II}\) type recurrence, generalized eigenvalue problem and orthogonal polynomials on the unit circle. Linear Algebra Appl. 562, 63–90 (2019)

  17. 17.

    Martínez-Finkelshtein, A., Silva Ribeiro, L.L., Sri Ranga, A., Tyaglov, M.: Complementary Romanovski–Routh polynomials: from orthogonal polynomials on the unit circle to Coulomb wave functions. Proc. Am. Math. Soc. 147, 2625–2640 (2019)

  18. 18.

    Meligy, A.S.: Simple expansion for the regular Coulomb wave function. Nuclear Phys. 6, 440–442 (1958)

  19. 19.

    Michel, N.: Precise Coulomb wave functions for a wide range of complex \(\ell \), \(\eta \) and \(z\). Comput. Phys. Commun. 176, 232–249 (2007)

  20. 20.

    Miyazaki, Y., Kikuchi, Y., Cai, D., Ikebe, Y.: Error analysis for the computation of zeros of regular Coulomb wave function and its first derivative. Math. Comp. 70, 1195–1204 (2001)

  21. 21.

    Neuman, E.: On Hahn polynomials and continuous dual Hahn polynomials. J. Comput. Anal. Appl. 8, 229–248 (2006)

  22. 22.

    Nikiforov, A.F., Uvarov, V.B.: Special Functions of Mathematical Physics: A Unified Introduction with Applications. Birkhäuser Verlag, Basel, Translated from Russian by R.P. Boas (1988)

  23. 23.

    Powel, J.L.: Recurrence formulas for Coulomb wave functions. Phys. Rev. 72, 626–627 (1947)

  24. 24.

    Rainville, E.D.: Special Functions. MacMillan, New York (1960)

  25. 25.

    Raposo, A.P., Weber, H.J., Alvarez-Castillo, D.E., Kirchbach, M.: Romanovski polynomials in selected physics problems. Cent. Eur. J. Phys. 5, 253–284 (2007)

  26. 26.

    Romanovski, V.: Sur quelques classes nouvelles de polynomes orthogonaux. C. R. Acad. Sci. Paris 188, 1023–1025 (1929)

  27. 27.

    Routh, E.J.: On some properties of certain solutions of a differential equation of the second order. Proc. Lond. Math. Soc. 16, 245–261 (1884)

  28. 28.

    Simon, B.: Orthogonal Polynomials on the Unit Circle. Part 1. Classical Theory, Amer. Math. Soc. Colloq. Publ., vol. 54, part 1, Amer. Math. Soc., Providence, RI (2005)

  29. 29.

    Štampach, F., Šťovíček, P.: Orthogonal polynomials associated with Coulomb wave functions. J. Math. Anal. Appl. 419, 231–254 (2015)

  30. 30.

    Sri Ranga, A.: Szegő polynomials from hypergeometric functions. Proc. Am. Math. Soc. 138, 4243–4247 (2010)

  31. 31.

    Slater, L.J.: Confluent Hypergeometric Functions. Cambridge University Press, Cambridge (1960)

  32. 32.

    Shepanski, J.R., Butler, S.T.: An expansion for Coulomb wave functions. Nuclear Phys. 1, 313–321 (1956)

  33. 33.

    Szegö, G.:, Orthogonal Polynomials. 4th ed., Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, RI (1975)

  34. 34.

    Thompson, I.J., Barnett, A.R.: Coulomb and Bessel functions of complex arguments and order. J. Comput. Phys. 64, 490–509 (1986)

  35. 35.

    Weber, H.J.: Connection between Romanovski polynomials and other polynomials. Centr. Eur. J. Math. 5, 581–595 (2007)

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Acknowledgements

The colaboration of A. Martínez-Finkelshtein 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. The author L.L. Silva Ribeiro was supported by the Grant 2017/04358-8 from Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) of Brazil. The work of A. Sri Ranga was supported by the Grants 2016/09906-0 of Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) and 304087/2018-1 of Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) of Brazil. The author M. Tyaglov was partially supported by The Program for Professor of Special Appointment (Oriental Scholar) at Shanghai Institutions of Higher Learning, by the Joint NSFC-ISF Research Program, jointly funded by the National Natural Science Foundation of China and the Israel Science Foundation (No.11561141001), and by National Natural Science Foundation of China under Grant No. 11871336.

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Correspondence to L. L. Silva Ribeiro.

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Martínez-Finkelshtein, A., Silva Ribeiro, L.L., Sri Ranga, A. et al. Complementary Romanovski–Routh Polynomials, Orthogonal Polynomials on the Unit Circle, and Extended Coulomb Wave Functions. Results Math 75, 42 (2020). https://doi.org/10.1007/s00025-020-1167-8

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Keywords

  • Romanovski–Routh polynomials
  • second order differential equations
  • orthogonal polynomials on the unit circle
  • para-orthogonal polynomials

Mathematics Subject Classification

  • 42C05
  • 33C47