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Biorthogonality and para-orthogonality of \(R_I\) polynomials

  • Kiran Kumar Behera
  • A. Swaminathan
Article
  • 80 Downloads

Abstract

In this paper, a sequence of linear combination of \(R_{I}\) polynomials such that the terms in this sequence have a common zero is constructed. A biorthogonality relation arising from such a sequence is discussed. Besides, a sequence of para-orthogonal polynomials by removing the common zero using suitable conditions is obtained. Finally, a case of hypergeometric functions is studied with numerical observations to illustrate the results obtained.

Keywords

\(R_{I}\) recurrence relations Linear combinations of polynomials Biorthogonality Para-orthogonal polynomials 

Mathematics Subject Classification

Primary 42C05 15A18 33C45 

Notes

Acknowledgements

The authors wish to thank the anonymous referees for their constructive criticism of the work which has helped in removing many ambiguities in the text and in particular for the numerical part in Sect. 5.

References

  1. 1.
    Alfaro, M., Marcellán, F., Peña, A., Rezola, M.L.: When do linear combinations of orthogonal polynomials yield new sequences of orthogonal polynomials? J. Comput. Appl. Math. 233(6), 1446–1452 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Askey, R.: Discussion of Szegö’s paper “Beiträge zur Theorie der Toeplitzschen Formen”. In: Askey, R. (ed.) Gabor Szegö. Collected works, vol. I, pp. 303–305. Birkhäuser, Boston, MA (1982)CrossRefGoogle Scholar
  3. 3.
    Askey, R.: Some problems about special functions and computations. Rend. Sem. Mat. Univ. Politec. Torino 1985, Special Issue, 1–22Google Scholar
  4. 4.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Brezinski, C., Driver, K.A., Redivo-Zaglia, M.: Quasi-orthogonality with applications to some families of classical orthogonal polynomials. Appl. Numer. Math. 48(2), 157–168 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Castillo, K., Costa, M.S., Sri Ranga, A., Veronese, D.O.: A Favard type theorem for orthogonal polynomials on the unit circle from a three term recurrence formula. J. Approx. Theory 184, 146–162 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Costa, M.S., Felix, H.M., Sri Ranga, A.: Orthogonal polynomials on the unit circle and chain sequences. J. Approx. Theory 173, 14–32 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Chihara, T.S.: On quasi-orthogonal polynomials. Proc. Am. Math. Soc. 8, 765–767 (1957)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Delsarte, P., Genin, Y.V.: The split Levinson algorithm. IEEE Trans. Acoust. Speech Signal Process. 34(3), 470–478 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Dickinson, D.: On quasi-orthogonal polynomials. Proc. Am. Math. Soc. 12, 185–194 (1961)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Draux, A.: On quasi-orthogonal polynomials of order \(r\). Integral Transforms Spec. Funct. 27(9), 747–765 (2016)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Driver, K., Muldoon, M.E.: Common and interlacing zeros of families of Laguerre polynomials. J. Approx. Theory 193, 89–98 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Fejér, L.: Mechanische Quadraturen mit positiven Cotesschen Zahlen. Math. Z. 37, 287–309 (1933)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Gibson, P.C.: Common zeros of two polynomials in an orthogonal sequence. J. Approx. Theory 105(1), 129–132 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Hendriksen, E., Njåstad, O.: Biorthogonal Laurent polynomials with biorthogonal derivatives. Rocky Mt. J. Math. 21(1), 301–317 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Ismail, M.E.H.: Classical and Quantum Orthogonal Polynomials in One Variable, reprint of the 2005 original, Encyclopedia of Mathematics and its Applications, 98. Cambridge University Press, Cambridge (2009)Google Scholar
  17. 17.
    Ismail, M.E.H., Masson, D.R.: Generalized orthogonality and continued fractions. J. Approx. Theory 83(1), 1–40 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Jones, W.B., Njåstad, O., Thron, W.J.: Moment theory, orthogonal polynomials, quadrature, and continued fractions associated with the unit circle. Bull. Lond. Math. Soc. 21(2), 113–152 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Jones, W.B., Thron, W.J., Waadeland, H.: A strong Stieltjes moment problem. Trans. Am. Math. Soc. 261(2), 503–528 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Jordaan, K., Toókos, F.: Mixed recurrence relations and interlacing of the zeros of some \(q\)-orthogonal polynomials from different sequences. Acta Math. Hungar. 128(1–2), 150–164 (2010)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Marcellán, F., Peherstorfer, F., Steinbauer, R.: Orthogonality properties of linear combinations of orthogonal polynomials. Adv. Comput. Math. 5(4), 281–295 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Riesz, M.: Sur le problème des moments, Troisième Note. Ark. Mat. Fys. 17, 1–52 (1923)Google Scholar
  23. 23.
    Shohat, J.: On mechanical quadratures, in particular, with positive coefficients. Trans. Am. Math. Soc. 42(3), 461–496 (1937)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    da Silva, A.P., Sri Ranga, A.: Polynomials generated by a three term recurrence relation: bounds for complex zeros. Linear Algebra Appl. 397, 299–324 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Simon, B.: Orthogonal Polynomials on the Unit Circle. Part 1, American Mathematical Society Colloquium Publications, vol. 54. American Mathematical Society, Providence, RI (2005)Google Scholar
  26. 26.
    Sri Ranga, A.: Szegő polynomials from hypergeometric functions. Proc. Am. Math. Soc. 138(12), 4259–4270 (2010)zbMATHCrossRefGoogle Scholar
  27. 27.
    Szegö, G.: Orthogonal Polynomials, American Mathematical Society Colloquium Publications, vol. 23, 4th edn. American Mathematical Society, Providence, RI (1975)Google Scholar
  28. 28.
    Tcheutia, D.D., Jooste, A.S., Koepf, W.: Mixed recurrence equations and interlacing properties for zeros of sequences of classical \(q\)-orthogonal polynomials. Appl. Numer. Math. 125, 86–102 (2018)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Temme, N.M.: Uniform asymptotic expansion for a class of polynomials biorthogonal on the unit circle. Constr. Approx. 2(4), 369–376 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Wong, M.L.: First and second kind paraorthogonal polynomials and their zeros. J. Approx. Theory 146(2), 282–293 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Zhedanov, A.: On the polynomials orthogonal on regular polygons. J. Approx. Theory 97(1), 1–14 (1999)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Istituto di Informatica e Telematica del Consiglio Nazionale delle Ricerche 2018

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology, RoorkeeRoorkeeIndia

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