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Ladder operators for Szego polynomials and related biorthogonal rational functions


Authors: Mourad E. H. Ismail and Mizan Rahman
Journal: Proc. Amer. Math. Soc. 124 (1996), 2149-2159
MSC (1991): Primary 33D45; Secondary 30E05
DOI: https://doi.org/10.1090/S0002-9939-96-03304-7
MathSciNet review: 1350949
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Abstract | References | Similar Articles | Additional Information

Abstract: We find the raising and lowering operators for orthogonal polynomials on the unit circle introduced by Szego and for their four parameter generalization to ${}_4\phi _3$ biorthogonal rational functions on the unit circle.


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

  • 1. W. A. Al-Salam and T. S. Chihara, Convolutions of orthogonal polynomials, SIAM J. Math. Anal. 7 (1976), 16--28. MR 53:3381
  • 2. W. A. Al-Salam and M. E. H. Ismail, A q-beta integral on the unit circle and some biorthogonal rational functions, Proc. Amer. Math. Soc. 121 (1994), 553--561. MR 94h:33011
  • 3. W. A. Al-Salam and A. Verma, $Q$-analogs of some biorthogonal functions, Canad. Math. Bull. 26(1983), 225--227. MR 84e:33010
  • 4. G. E. Andrews and R. A. Askey, Classical orthogonal polynomials, in ``Polynomes Orthogonaux et Applications", eds. C. Breziniski et ál., Lecture Notes in Mathematics, vol. 1171, Springer-Verlag, Berlin, 1984, pp. 36-63.
  • 5. R. A. Askey and M. E. H. Ismail, A generalization of ultraspherical polynomials, in ``Studies in Pure Mathematics", ed. P. Erdös, Birkhauser, Basel, 1983, pp. 55-78. MR 87a:33015
  • 6. R. A. Askey and J. A. Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Memoirs Amer. Math. Soc. Number 319 (1985). MR 87a:05023
  • 7. C. Berg and M. E. H. Ismail, $Q$-Hermite polynomials and classical orthogonal polynomials, Canad. J. Math. (1996), to appear.
  • 8. W. D. Evans, B. M. Brown and M. E. H. Ismail, The Askey-Wilson polynomials and $q$-Sturm-Liouville problems, Math. Proc. Camb. Philos. Soc. 119 (1996), 1--16.
  • 9. G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge University Press, Cambridge, 1990. MR 91d:33034
  • 10. U. Grenander and G. Szego, Toeplitz Forms and Their Applications, University of California Press, Berkely, 1958, reprinted by Chelsea, Bronx, 1984. MR 88b:42031
  • 11. M. E. H. Ismail and D. R. Masson, Q-Hermite polynomials, biorthogonal rational functions, Transactions Amer. Math. Soc. 346 (1994), 63--110. CMP 94:16
  • 12. P. I. Pastro, Orthogonal polynomials and some $q$-beta integrals of Ramanujan, J. Math. Anal. Appl. 112 (1985), 517-540. MR 87c:33015
  • 13. M. Rahman, Biorthogonality of a system of rational functions with respect to a positive measure on $[-1,1]$, SIAM J. Math. Anal. 22 (1991), 1421-1431. MR 92h:33016
  • 14. M. Rahman and S. K. Suslov, Classical biorthogonal rational functions in "Methods of Approximation Theory in Complex Analysis and Mathematical Physics", A. A. Goncar and E. B. Saff, editors, Lecture Notes in Mathematics 1550, Springer-Verlag, Berlin, pp. 131-150.
  • 15. G. Szego, Beitrag zur Theorie der Thetafunktionen, Sitz. Preuss. Akad. Wiss. Phys. Math. Kl., XIX (1926), 242-252, Reprinted in "Collected Papers", edited by R. Askey, Volume I, Birkhauser, Boston, 1982.

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Additional Information

Mourad E. H. Ismail
Affiliation: Department of Mathematics, University of South Florida, Tampa, Florida 33620

Mizan Rahman
Affiliation: Department of Mathematics, Carleton University, Ottawa, Ontario, Canada K1S 5B6

DOI: https://doi.org/10.1090/S0002-9939-96-03304-7
Keywords: Szeg\H{o} polynomials, $q$-difference operators, orthogonality on the unit circle, $q$-beta integrals, biorthogonal rational functions, raising and lowering operators, $q$-Sturm-Liouville equations.
Received by editor(s): July 5, 1994
Received by editor(s) in revised form: February 2, 1995
Additional Notes: Research partially supported by NSF grant DMS 9203659 and NSERC grant A6197
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1996 American Mathematical Society

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