Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Szegő polynomials from hypergeometric functions


Author: A. Sri Ranga
Journal: Proc. Amer. Math. Soc. 138 (2010), 4259-4270
MSC (2010): Primary 33C05, 42C05; Secondary 33C45
DOI: https://doi.org/10.1090/S0002-9939-2010-10592-0
Published electronically: July 30, 2010
MathSciNet review: 2680052
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Szegő polynomials with respect to the weight function $ \omega(\theta) = e^{\eta \theta} [\sin(\theta/2)]^{2\lambda}$, where $ \eta, \lambda \in \mathbb{R}$ and $ \lambda > -1/2$ are considered. Many of the basic relations associated with these polynomials are given explicitly. Two sequences of para-orthogonal polynomials with explicit relations are also given.


References [Enhancements On Off] (What's this?)

  • 1. G.E. Andrews, R. Askey and R. Roy, ``Special Functions'', Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2000. MR 1688958 (2000g:33001)
  • 2. R. Askey (editor), ``Gabor Szegő: Collected Papers. Volume 1'', Contemporary Mathematics, Amer. Math. Soc., Providence, RI, 1982. MR 674482 (84d:01082a)
  • 3. A. Cachafeiro, F. Marcellán and C. Pérez, Orthogonal polynomials with respect to the sum of an arbitrary measure and a Bernstein-Szegö measure, Adv. Comput. Math., 26 (2007), 81-104. MR 2350346 (2008m:33032)
  • 4. R. Cruz-Barroso, P. González-Vera and F. Perdomo-Pĭo, Quadrature formulas associated with Rogers-Szegő polynomials, Comput. Math. Appl., 57 (2009), 308-323. MR 2488385 (2009k:65040)
  • 5. L. Daruis, O. Njastad, W. Van Assche, Szegő quadrature and frequency analysis, Electron. Trans. Numer. Anal., 19 (2005), 48-57. MR 2149269 (2006e:41057)
  • 6. G. Gasper, Orthogonality of certain functions with complex valued weights, Canad. J. Math., 33 (1981), 1261-1270. MR 638380 (83a:33014)
  • 7. Ya.L. Geronimus, ``Orthogonal Polynomials'', Amer. Math. Soc. Transl., Ser. 2, vol. 108, American Mathematical Society, Providence, RI, 1977.
  • 8. L. Golinskii and A. Zlatoš, Coefficients of orthogonal polynomials on the unit circle and higher-order Szegő theorems, Constr. Approx., 26 (2007), 361-382. MR 2335688 (2008k:42080)
  • 9. E. Hendriksen and H. van Rossum, Orthogonal Laurent polynomials, Indag. Math. (ser. A), 48 (1986), 17-36. MR 834317 (87j:30008)
  • 10. M.E.H. Ismail, ``Classical and Quantum Orthogonal Polynomials in One Variable'', Encyclopedia of Mathematics and Its Applications, vol. 98, Cambridge Univ. Press, Cambridge, UK, 2005. MR 2191786 (2007f:33001)
  • 11. W.B. Jones, O. Njåstad and W.J. Thron, Moment theory, orthogonal polynomials, quadrature, and continued fractions associated with the unit circle, Bull. London Math. Soc., 21 (1989), 113-152. MR 976057 (90e:42027)
  • 12. W.B. Jones and W.J. Thron, ``Continued Fractions Analytic Theory and Applications'', Encyclopedia of Mathematics and Its Applications, vol. 11, Addison-Wesley, Reading, MA, 1980. MR 595864 (82c:30001)
  • 13. L. Lorentzen and H. Waadeland, ``Continued Fractions with Applications'', Studies in Computational Mathematics, vol. 3, North-Holland, Amsterdam, 1992. MR 1172520 (93g:30007)
  • 14. A.L. Lukashov and F. Peherstorfer, Zeros of polynomials orthogonal on two arcs of the unit circle, J. Approx. Theory, 132 (2005), 42-71. MR 2110575 (2006g:42045)
  • 15. A. Martínez-Finkelshtein, K.T.-R. McLaughlin and E.B. Saff, Szegő orthogonal polynomials with respect to an analytic weight: canonical representation and strong asymptotics, Constr. Approx., 24 (2006), 319-363. MR 2253965 (2007e:42029)
  • 16. J. Petronilho, Orthogonal polynomials on the unit circle via a polynomial mapping on the real line, J. Comput. Appl. Math., 216 (2008), 98-127. MR 2421843 (2009e:42054)
  • 17. B. Simon, ``Orthogonal Polynomials on the Unit Circle. Part 1. Classical Theory'', American Mathematical Society Colloquium Publications, vol. 54, part 1, American Mathematical Society, Providence, RI, 2004. MR 2105088 (2006a:42002a)
  • 18. B. Simon, ``Orthogonal Polynomials on the Unit Circle. Part 2. Spectral Theory'', American Mathematical Society Colloquium Publications, vol. 54, part 2, American Mathematical Society, Providence, RI, 2004. MR 2105089 (2006a:42002b)
  • 19. B. Simon, Equilibrium measures and capacities in spectral theory, Inverse Probl. Imaging, 1 (2007), 713-772. MR 2350223 (2008k:31003)
  • 20. G. Szegő, Über Beiträge zur theorie der toeplitzschen formen, Math. Z., 6 (1920), 167-202. MR 1544404
  • 21. G. Szegő, Über Beiträge zur theorie der toeplitzschen formen, II, Math. Z., 9 (1921), 167-190. MR 1544462
  • 22. G. Szegő, ``Orthogonal Polynomials'', 4th ed., Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, RI, 1975. MR 0372517 (51:8724)
  • 23. N.M. Temme, Uniform asymptotic expansion for a class of polynomials biorthogonal on the unit circle, Constr. Approx., 2 (1986), 369-376. MR 892162 (88e:42047)
  • 24. S. Tsujimoto and A. Zhedanov, Elliptic hypergeometric Laurent biorthogonal polynomials with a dense point spectrum on the unit circle, SIGMA Symmetry Integrability Geom. Methods Appl., 5 (2009), Paper 033, 30 pp. MR 2506179 (2010g:33018)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 33C05, 42C05, 33C45

Retrieve articles in all journals with MSC (2010): 33C05, 42C05, 33C45


Additional Information

A. Sri Ranga
Affiliation: Departamento de Ciências de Computao e Estatística, Ibilce, Universidade Estadual Paulista, 15054-000, São José do Rio Preto, SP, Brazil
Email: ranga@ibilce.unesp.br

DOI: https://doi.org/10.1090/S0002-9939-2010-10592-0
Keywords: Hypergeometric function, continued fractions, Szegő polynomials
Received by editor(s): May 14, 2009
Received by editor(s) in revised form: November 3, 2009
Published electronically: July 30, 2010
Communicated by: Peter A. Clarkson
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society