Skip to main content
SpringerLink
Log in

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

  • Published:
Results in Mathematics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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. 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)

    Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

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

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  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)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  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)

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  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)

    Google Scholar 

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

    MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. Departament of Mathematics, Baylor University, Waco, TX,  76798-7328, USA

    A. Martínez-Finkelshtein

  2. Departamento de Matemática Aplicada, IBILCE, UNESP-Universidade Estadual Paulista, São José do Rio Preto, SP, Brazil

    L. L. Silva Ribeiro & A. Sri Ranga

  3. School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, China

    M. Tyaglov

Authors
  1. A. Martínez-Finkelshtein
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. L. Silva Ribeiro
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. Sri Ranga
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. M. Tyaglov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. L. Silva Ribeiro.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00025-020-1167-8

Keywords

Mathematics Subject Classification

Search

Navigation