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 biorthogonal rational functions on the unit circle.

**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,*-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,*-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 -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 -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*, 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