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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
MathSciNet review:
1350949
Full-text PDF Free Access
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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  biorthogonal rational functions on the unit circle.
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- 2.
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- 3.
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 -analogs of some biorthogonal functions, Canad. Math. Bull. 26(1983), 225--227. MR 84e:33010
- 4.
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- 5.
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- P. I. Pastro, Orthogonal polynomials and some
 -beta integrals of Ramanujan, J. Math. Anal. Appl. 112 (1985), 517-540. MR 87c:33015
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- M. Rahman, Biorthogonality of a system of rational functions with respect to a positive measure on
![$[-1,1]$](/proc/1996-124-07/S0002-9939-96-03304-7/gif-references/img241.gif) , SIAM J. Math. Anal. 22 (1991), 1421-1431. MR 92h:33016
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- 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:
http://dx.doi.org/10.1090/S0002-9939-96-03304-7
PII:
S 0002-9939(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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