Ladder operators for Szegö polynomials and related biorthogonal rational functions

Ismail, Mourad; Rahman, Mizan

doi:10.1090/S0002-9939-96-03304-7

Ladder operators for Szegö polynomials and related biorthogonal rational functions
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by Mourad E. H. Ismail and Mizan Rahman

Proc. Amer. Math. Soc. 124 (1996), 2149-2159

DOI: https://doi.org/10.1090/S0002-9939-96-03304-7

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Abstract:

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

References

W. A. Al-Salam and T. S. Chihara, Convolutions of orthonormal polynomials, SIAM J. Math. Anal. 7 (1976), no. 1, 16–28. MR 399537, DOI 10.1137/0507003
Waleed A. Al-Salam and Mourad E. H. Ismail, A $q$-beta integral on the unit circle and some biorthogonal rational functions, Proc. Amer. Math. Soc. 121 (1994), no. 2, 553–561. MR 1215197, DOI 10.1090/S0002-9939-1994-1215197-0
W. A. Al-Salam and A. Verma, $q$-analogs of some biorthogonal functions, Canad. Math. Bull. 26 (1983), no. 2, 225–227. MR 697805, DOI 10.4153/CMB-1983-035-8
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.
R. Askey and Mourad E. H. Ismail, A generalization of ultraspherical polynomials, Studies in pure mathematics, Birkhäuser, Basel, 1983, pp. 55–78. MR 820210
Richard Askey and James Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54 (1985), no. 319, iv+55. MR 783216, DOI 10.1090/memo/0319
C. Berg and M. E. H. Ismail, $Q$-Hermite polynomials and classical orthogonal polynomials, Canad. J. Math. (1996), to appear.
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.
George Gasper and Mizan Rahman, Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 35, Cambridge University Press, Cambridge, 1990. With a foreword by Richard Askey. MR 1052153
Ulf Grenander and Gábor Szegő, Toeplitz forms and their applications, 2nd ed., Chelsea Publishing Co., New York, 1984. MR 890515
M. E. H. Ismail and D. R. Masson, Q-Hermite polynomials, biorthogonal rational functions, Transactions Amer. Math. Soc. 346 (1994), 63–110.
P. I. Pastro, Orthogonal polynomials and some $q$-beta integrals of Ramanujan, J. Math. Anal. Appl. 112 (1985), no. 2, 517–540. MR 813618, DOI 10.1016/0022-247X(85)90261-6
Mizan Rahman, Biorthogonality of a system of rational functions with respect to a positive measure on $[-1,1]$, SIAM J. Math. Anal. 22 (1991), no. 5, 1430–1441. MR 1112517, DOI 10.1137/0522091
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.
G. Szegő, 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.