Discriminants and functional equations for polynomials orthogonal on the unit circle

Journal of Approximation Theory

Volumn 110, Issue 2, 2001, Pages 200-228

  Ismail, Mourad E H ^a   Witte, Nicholas S ^b  

^a UNIVERSITY OF SOUTH FLORIDA   (United States)

^b UNIVERSITY OF MELBOURNE   (Australia)

Author keywords

Discriminants; polynomials orthogonal on the unit circle; differential equations; zeros

Indexed keywords

EID: 0037727538     PISSN: 00219045     EISSN: None     Source Type: Journal    
DOI: 10.1006/jath.2000.3540     Document Type: Article

References (35)

1. Andrews G. E., Askey R. A., Roy R. Special Functions. 1999;Cambridge University Press, Cambridge.
2. Bauldry W. Estimates of asymmetric Freud polynomials on the real line. J. Approx. Theory. 63:1990;225-237.
3. Baik J., Deift P., Johansson K. On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12:1999;1119-1178.
4. Bonan S. S., Clark D. S. Estimates of the Hermite and the Freud polynomials. J. Approx. Theory. 63:1990;210-224.
5. Chen Y., Ismail M. E. H. Ladder operators and differential equations for orthogonal polynomials. J. Phys. A. 30:1997;7818-7829.
6. Dickson L. E. New Course on the Theory of Equations. 1939;Wiley, New York.
7. Forrester P. J., Rogers J. B. Electrostatics and the zeros of the classical orthogonal polynomials. SIAM J. Math. Anal. 17:1986;461-468.
8. Freud G. Orthogonal Polynomials. 1971;Pergamon, Oxford.
9. Gasper G., Rahman M. Basic Hypergeometric Series. 1990;Cambridge University Press, Cambridge.
10. Geronimus L. Ya. Orthogonal Polynomials. 1971;Consultants Bureau, New York.
11. Geronimus L. Ya. Polynomials orthogonal on a circle and their applications. Amer. Math. Soc. Transl. 3:1962;Amer. Math. Soc. p. 1-78.
12. Geronimus L. Ya. Orthogonal Polynomials. Amer. Math. Soc. Transl. 108:1977;Amer. Math. Soc. p. 3-130.
13. Gessel I. M. Symmetric functions and P-recursiveness. J. Comput. Theor., Ser. A. 53:1990;257-285.
14. Grunbaum F. A. Variation on a theme of Stieltjes and Heine: an electrostatic interpretation of zeros of certain polynomials. J. Comput. Appl. Math. 99:1998;189-194.
15. Hilbert D. Über die Discriminante der in Endlichen abbrechenden hypergeometrischen Reihe. J. Reine Angewandte Mat. 103:1885;337-345.
16. Hisakado M. Unitary matrix models and painléve III. Mod. Phys. Lett. A. 11:1996;3001-3010.
17. Horn R. A., Johnson C. Matrix Analysis. 1992;Cambridge University Press, Cambridge.
18. Ismail M. E. H. Discriminants and functions of the second kind of orthogonal polynomials. Results Math. 34:1998;132-149.
19. Ismail M. E. H. An electrostatics model for zeros of general orthogonal polynomials. Pacific J. Math. 193:2000;355-369.
20. Ismail M. E. H. More on electrostatics models for zeros of orthogonal polynomials. J. Nonlinear Funct. Anal. Optim. 21:2000;191-204.
21. Ismail M. E. H. Difference equations and quantized discriminants for q-orthogonal polynomials. Adv. in Appl. Math. 2000.
22. β+Mδ(x+1)+Nδ(x-1). Canad. Math Bull. 27:1984;205-214.
23. A. Magnus, MAPA3072A Special topics in approximation theory 1999-2000: Semi-classical orthogonal polynomials on the unit circle, available at, http://www.math.ucl.ac.be/magnus/ .
24. Nevai P. Géza Freud, orthogonal polynomials, and Christoffel functions. A case study. J. Approx. Theory. 48:1986;3-167.
25. Pastro P. I. Orthogonal polynomials and some q-beta integrals of Ramanujan. J. Math. Anal. Appl. 112:1985;517-540.
26. Periwal V., Shevitz D. Unitary-matrix models as exactly solvable string theories. Phys. Rev. Lett. 64:1990;1326-1329.
27. Saff E. B., Totik V. Logarithmic Potentials with External Fields. 1997;Springer-Verlag, New York.
28. Schur I. Affektlose Gleichungen in der Theorie der Laguerreschen und Hermiteschen Polynomes. J. Reine Angew. Mat. 165:1931;52-58.
29. Selberg A. Bemerkninger om et multiplet integral. Norsk Mat. Tidsskr. 26:1944;71-78.
30. Stieltjes T. J. Sur quelques théorèmes d'algèbre. C.R. Acad. Sci. Paris. 100:1885;439 - 440.
31. Stieltjes T. J. Sur les polynômes de Jacobi. C.R. Acad. Sci. Paris. 100:1885;620-622.
32. Szego G. Ein Beitrag zur Theorie der Thetafunktionen. Sitz. Preuss. Akad. Wiss. Phys. Math. Kl. 19:1926;242-252.
33. Szego G. Orthogonal Polynomials. 1975;Amer. Math. Soc. Providence.
34. Tracy C. A., Widom H. Random unitary matrices, permutations and Painlevé Commun. Math. Phys. 207:1999;665-685.
35. Witte N. S., Forrester P. J. Gap probabilities in the finite and scaled Cauchy random matrix ensembles. Nonlinearity. 13:2000;1965-1986.