Skip to main content
SpringerLink
Log in

Matrix Gegenbauer Polynomials: The \(2\times 2\) Fundamental Cases

  • Published:
Constructive Approximation Aims and scope

Abstract

In this paper, we exhibit explicitly a sequence of \(2 \times 2\) matrix valued orthogonal polynomials with respect to a weight \(W_{p,n}\), for any pair of real numbers p and n such that \(0<p<n\). The entries of these polynomiales are expressed in terms of the Gegenbauer polynomials \(C_k^\lambda \). The corresponding three-term recursion relations are also given, and we make some studies of the algebra of differential operators associated with the weight \(W_{p,n}\).

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. Castro, M.M., Grünbaum, F.A.: Orthogonal matrix polynomials satisfying first order differential equations: a collection of instructive examples. J. Nonlinear Math. Phys. 12(2), 63–67 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Castro, M.M., Grünbaum, F.A.: The algebra of differential operators associated to a given family of matrix valued orthogonal polynomials: five instructive examples. Int. Math. Res. Not. 27(2), 1–33 (2006)

    MATH  Google Scholar 

  3. Durán, A.J.: Matrix inner product having a matrix symmetric second-order differential operator. Rocky Mt. J. Math. 27(2), 585–600 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Durán, A.J., Grünbaum, F.A.: Orthogonal matrix polynomials satisfying second-order differential equations. Int. Math. Res. Not. 10, 461–484 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Durán, A.J., Grünbaum, F.A.: A characterization for a class of weight matrices with orthogonal matrix polynomials satisfying second-order differential equations. Int. Math. Res. Not. 23, 1371–1390 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Durán, A.J., Grünbaum, F.A.: Structural formulas for orthogonal matrix polynomials satisfying second-order differential equations. I. Constr. Approx. 22(2), 255–271 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Grünbaum, F.A.: Matrix valued Jacobi polynomials. Bull. Sci. Math. 127(3), 207–214 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Grünbaum, F.A., Pacharoni, I., Tirao, J.: A matrix-valued solution to Bochner’s problem. J. Phys. A 34(48), 10647–10656 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Grünbaum, F.A., Pacharoni, I., Tirao, J.: Matrix valued spherical functions associated to the complex projective plane. J. Funct. Anal. 188(2), 350–441 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Grünbaum, F.A., Pacharoni, I., Tirao, J.: Matrix valued orthogonal polynomials of the Jacobi type. Indag. Math. (N.S.) 14(3–4), 353–366 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. Grünbaum, F.A., Pacharoni, I., Tirao, J.: Matrix valued orthogonal polynomials of Jacobi type: the role of group representation theory. Ann. Inst. Fourier (Grenoble) 55(6), 2051–2068 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Grünbaum, F.A., Pacharoni, I., Zurrián, I.N.: Time and band limiting for matrix valued functions, an example. SIGMA (2015). doi:10.3842/SIGMA.2015.044

  13. Grünbaum, F.A., Tirao, J.: The algebra of differential operators associated to a weight matrix. Integral Equ. Oper. Theory 58(4), 449–475 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Koekoek, R., Swarttouw, R.F.: The Askey-scheme of hypergeometric orthogonal polynomials and its \(q\)-analogue. Faculty of Technical Mathematics and Informatics Report 98–17, Technische Universiteit Delft, Delft, Netherlands (1998)

  15. Koelink, E., De Los Ríos, A., Román, P.: Matrix-valued Gegenbauer polynomials. Preprint, arXiv:1403.293 (2014)

  16. Koelink, E., van Pruijssen, M., Román, P.: Matrix-valued orthogonal polynomials related to (SU(2) \(\times \) SU(2), diag). Int. Math. Res. Not. 2012(24), 5673–5730 (2012)

    MathSciNet  MATH  Google Scholar 

  17. Koelink, E., van Pruijssen, M., Román, P.: Matrix-valued orthogonal polynomials related to (SU(2) \(\times \) SU(2), SU(2)), II. PRIMS 49(2), 271–312 (2013)

    MATH  Google Scholar 

  18. Krein, M.G.: Infinite J-matrices and a matrix moment problem. Dokl. Akad. Nauk SSSR 69(2), 125–128 (1949)

    MathSciNet  Google Scholar 

  19. Krein, M.G.: Fundamental aspects of the representation theory of Hermitian operators with deficiency index \((m, m)\). AMS Trans. 2(97), 75–143 (1971)

    MATH  Google Scholar 

  20. Miranian, L.: On classical orthogonal polynomials and differential operators. J. Phys. A: Math. Gen. 38, 6379–6383 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Pacharoni, I., Román, P.: A sequence of matrix valued orthogonal polynomials associated to spherical functions. Constr. Approx. 28(2), 127–147 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Pacharoni, I., Tirao, J.: Three term recursion relation for spherical functions associated to the complex projective plane. Math. Phys. Anal. Geom. 7(3), 193–221 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. Pacharoni, I., Tirao, J.: Matrix valued orthogonal polynomials arising from the complex projective space. Constr. Approx. 25(2), 177–192 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Pacharoni, I., Tirao, J., Zurrián, I.: Spherical functions associated to the three dimensional sphere. Ann. Mat. Pura Appl. 193(6), 1727–1778 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. Szegö, G.: Orthogonal Polynomials, 4th edn. American Mathematical Soc, Providence (1975)

    MATH  Google Scholar 

  26. Tirao, J.: The algebra of differential operators associated to a weight matrix: a first example. In: Polcino Milies, C. (ed.) Groups, algebras and applications. XVIII Latin American algebra colloquium, São Pedro, Brazil, August 3–8, 2009. Proceedings, Contemporary Mathematics, vol. 537, pp. 291–324. American Mathematical Society (AMS), Providence (2011)

  27. Tirao, J., Zurrián, I.: Spherical functions in \({S}^n\), the fundamental types. S.I.G.M.A. 10, 41 (2014)

    MATH  Google Scholar 

  28. Zurrián, I.: The algebra of differential operators for a Gegenbauer weight matrix. Preprint, arXiv:1505.03321 (2015)

  29. Zurrián, I.: Funciones Esféricas Matriciales Asociadas a las Esferas y a los Espacios Proyectivos Reales. Tesis Doctoral. Universidad Nacional de Córdoba, 2013. http://www2.famaf.unc.edu.ar/publicaciones/documents/serie_d/DMat76. arXiv:1306.6581

Download references

Acknowledgments

We would like to thank the referees for many useful comments and suggestions that helped us to improve a first version of this paper.

Author information

Authors and Affiliations

  1. CIEM-FaMAF, Universidad Nacional de Córdoba, 5000, Córdoba, Argentina

    Inés Pacharoni & Ignacio Zurrián

Authors
  1. Inés Pacharoni
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ignacio Zurrián
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Inés Pacharoni.

Additional information

Communicated by Erik Koelink.

This paper was partially supported by CONICET, PIP 112-200801-01533 and SeCyT-UNC.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pacharoni, I., Zurrián, I. Matrix Gegenbauer Polynomials: The \(2\times 2\) Fundamental Cases. Constr Approx 43, 253–271 (2016). https://doi.org/10.1007/s00365-015-9301-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00365-015-9301-7

Keywords

Mathematics Subject Classification

Search

Navigation