Acta Applicandae Mathematicae

, Volume 121, Issue 1, pp 105–135 | Cite as

Higher Order Coherent Pairs

Article

Abstract

In this paper, we study necessary and sufficient conditions for the relation
$$\begin{array}{@{}l}P_n^{{[r]}}(x) + a_{n-1,r} P_{n-1}^{{[r]}}(x)= R_{n-r}(x) + b_{n-1,r} R_{n-r-1}(x),\\[5pt]\quad a_{n-1,r}\neq0,\ n\geq r+1,\end{array}$$
where {Pn(x)}n≥0 and {Rn(x)}n≥0 are two sequences of monic orthogonal polynomials with respect to the quasi-definite linear functionals \(\mathcal{U},\mathcal{V}\), respectively, or associated with two positive Borel measures μ0,μ1 supported on the real line. We deduce the connection with Sobolev orthogonal polynomials, the relations between these functionals as well as their corresponding formal Stieltjes series. As sake of example, we find the coherent pairs when one of the linear functionals is classical.

Keywords

Coherent pairs Sobolev inner product Stieltjes functions Semiclassical linear functionals Orthogonal polynomials 

Mathematics Subject Classification (2000)

42C05 

References

  1. 1.
    Alfaro, M., Marcellán, F., Peña, A., Rezola, M.L.: On linearly related orthogonal polynomials and their functionals. J. Math. Anal. Appl. 287, 307–319 (2003) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Alfaro, M., Marcellán, F., Peña, A., Rezola, M.L.: On rational transformations of linear functionals: direct problem. J. Math. Anal. Appl. 298, 171–183 (2004) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    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, 1446–1452 (2010) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Alfaro, M., Marcellán, F., Peña, A., Rezola, M.L.: Orthogonal polynomials associated with an inverse quadratic spectral transform. Comput. Math. Appl. 61, 888–900 (2011) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Bonan, S., Nevai, P.: Orthogonal polynomials and their derivatives, I. J. Approx. Theory 40, 134–147 (1984) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Bonan, S., Lubinsky, D.S., Nevai, P.: Orthogonal polynomials and their derivatives, II. SIAM J. Math. Anal. 18(4), 1163–1176 (1987) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Branquinho, A., Rebocho, M.N.: On the semiclassical character of orthogonal polynomials satisfying structure relations. J. Differ. Equ. Appl. 18, 111–138 (2012) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Branquinho, A., Foulquié Moreno, A., Marcellán, F., Rebocho, M.N.: Coherent pairs of linear functionals on the unit circle. J. Approx. Theory 153, 122–137 (2008) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Chihara, T.S.: An Introduction to Orthogonal Polynomials. Gordon & Breach, New York (1978) MATHGoogle Scholar
  10. 10.
    de Jesus, M.N., Petronilho, J.: On linearly related sequences of derivatives of orthogonal polynomials. J. Math. Anal. Appl. 347, 482–492 (2008) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Delgado, A.M., Marcellán, F.: Companion linear functionals and Sobolev inner products: a case study. Methods Appl. Anal. 11(2), 237–266 (2004) MathSciNetMATHGoogle Scholar
  12. 12.
    Delgado, A.M., Marcellán, F.: On an extension of symmetric coherent pairs of orthogonal polynomials. J. Comput. Appl. Math. 178, 155–168 (2005) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Hahn, W.: Über die Jacobischen Polynome und Zwei Verwandte Polynomklassen. Math. Z. 39, 634–638 (1935) MathSciNetCrossRefGoogle Scholar
  14. 14.
    Hahn, W.: Über Höhere Ableitungen von Orthogonal Polynomen. Math. Z. 43, 101 (1937) CrossRefGoogle Scholar
  15. 15.
    Iserles, A., Koch, P.E., Nørsett, S.P., Sanz-Serna, J.M.: On polynomials orthogonal with respect to certain Sobolev inner products. J. Approx. Theory 65, 151–175 (1991) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Ismail, M.E.H.: Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia Math. Appl., vol. 98. Cambridge University Press, Cambridge (2005) MATHGoogle Scholar
  17. 17.
    Kwon, K.H., Lee, J.H., Marcellán, F.: Generalized coherent pairs. J. Math. Anal. Appl. 253, 482–514 (2001) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Marcellán, F., Petronilho, J.C.: Orthogonal polynomials and coherent pairs: the classical case. Indag. Math. 6(3), 287–307 (1995) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Marcellán, F., Branquinho, A., Petronilho, J.: Classical Orthogonal polynomials: a functional approach. Acta Appl. Math. 34, 283–303 (1994) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Marcellán, F., Pérez, T.E., Petronilho, J., Piñar, M.A.: What is beyond coherent pairs of orthogonal polynomials? J. Comput. Appl. Math. 65, 267–277 (1995) MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Marcellán, F., Pérez, T.E., Piñar, M.A.: Orthogonal polynomials on weighted Sobolev spaces: the semiclassical case. Ann. Numer. Math. 2, 93–122 (1995) MathSciNetMATHGoogle Scholar
  22. 22.
    Marcellán, F., Martínez-Finkelshtein, A., Moreno-Balcázar, J.: k-Coherence of measures with non-classical weights. In: Español, L., Varona, J.L. (eds.) Margarita Mathematica en Memoria de José Javier Guadalupe Hernández, pp. 77–83. Servicio de Publicaciones, Universidad de la Rioja, Logroño (2001) Google Scholar
  23. 23.
    Maroni, P.: Une théorie algébrique des polynômes orthogonaux. Application aux polynômes orthogonaux semi-classiques. In: Brezinski, C., Gori, L., Ronveaux, A. (eds.) Orthogonal Polynomials and Their Applications. IMACS Annals Comput. Appl. Math, vol. 9, pp. 95–130 (1991) Google Scholar
  24. 24.
    Maroni, P.: Semi-classical character and finite-type relations between polynomial sequences. Appl. Numer. Math. 31, 295–330 (1999) MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Maroni, P., Sfaxi, R.: Diagonal orthogonal polynomial sequences. Methods Appl. Anal. 7(4), 769–792 (2000) MathSciNetMATHGoogle Scholar
  26. 26.
    Martínez-Finkelshtein, A., Moreno-Balcázar, J.J., Pérez, T.E., Piñar, M.A.: Asymptotics of Sobolev orthogonal polynomials for coherent pairs of measures. J. Approx. Theory 92, 280–293 (1998) MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Meijer, H.G.: Coherent pairs and zeros of Sobolev-type orthogonal polynomials. Indag. Math. 4(2), 163–176 (1993) MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Meijer, H.G.: Determination of all coherent pairs. J. Approx. Theory 89, 321–343 (1997) MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Petronilho, J.: On the linear functionals associated to linearly related sequences of orthogonal polynomials. J. Math. Anal. Appl. 315, 379–393 (2006) MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Sfaxi, R., Alaya, J.: On orthogonal polynomials with respect to the form −(xc)S′. Period. Math. Hung. 52(1), 67–99 (2006) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad Carlos III de MadridLeganés, MadridSpain

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