q-Hermite polynomials and classical orthogonal polynomials

Canadian Journal of Mathematics

Volumn 48, Issue 1, 1996, Pages 43-63

  Berg, Christian ^a   Ismail, Mourad E H ^b  

^a UNIVERSITY OF COPENHAGEN   (Denmark)

^b UNIVERSITY OF SOUTH FLORIDA   (United States)

Author keywords

Askey Wilson polynomials; Biorthogonal rational functions; Orthogonality relations; q beta integrals; q Hermite polynomials; q orthogonal polynomials

Indexed keywords

EID: 0030543222     PISSN: 0008414X     EISSN: None     Source Type: Journal    
DOI: 10.4153/CJM-1996-002-4     Document Type: Article

References (28)

1. N. I. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis, English translation, Oliver and Boyd, Edinburgh, 1965.
2. W. A. Al-Salam and L. Carlitz, Some orthogonal q-polynomials, Math. Nachr. 30(1965), 47-61.
3. W. A. Al-Salam and T. S. Chihara, Convolutions of orthogonal polynomials, SIAM J. Math. Anal. 7(1976), 16-28.
4. 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.
5. W. A. Al-Salam and A. Verma, Q-analogs of some biorthogonal functions, Canad. Math. Bull. 26(1983), 225-227.
6. G. E. Andrews and R. A. Askey, Classical orthogonal polynomials. In: Polynomes Orthogonaux et Applications, (eds. C. Brezinski, et al), Lecture Notes in Math. 1171, Springer-Verlag, Berlin, 1984, 36-63.
7. R. A. Askey, Continuous q-Hermite polynomials when q > 1. In: q-Series and Partitions, (ed. D. Stanton), IMA Math. Appl., Springer-Verlag, New York, 1989, 151-158.
8. R. A. Askey and M. E. H. Ismail, A generalization of ultraspherical polynomials. In: Studies in Pure Math., (ed. P. Erdös), Birkhauser, Basel, 1983, 55-78.
9. _, Recurrence relations, continued fractions and orthogonal polynomials, Mem. Amer. Math. Soc. 300(1984).
10. R. A. Askey and J. A. Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 319(1985).
11. _, Associated Laguerre and Hermite polynomials, Proc. Royal Soc. Edinburgh Sect. A 96(1984), 15-37.
12. C. Berg and G. Valent, The Nevanlinna parameterization for some indeterminate Stieltjes moment problems associated with birth and death processes, Methods Appl. Anal. 1(1994), 169-209.
13. J. Bustoz and M. E. H. Ismail, The associated ultraspherical polynomials and their q-analogues, Canad. J. Math. 34(1982), 718-736.
14. T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978.
15. T. S. Chihara and M. E. H. Ismail, Extremal measures for a system of orthogonal polynomials, Constr. Approx. 9(1993), 111-119.
16. G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge Univ. Press, Cambridge, 1990.
17. M. E. H. Ismail, A queueing model and a set of orthogonal polynomials, J. Math. Anal. Appl. 108(1985), 575-594.
18. M. E. H. Ismail and D. R. Masson, Q-Hermite polynomials, biorthogonal rational functions and Q-beta integrals, Trans. Amer. Math. Soc. 346(1994), 61-116.
19. M. E. H. Ismail and M. Rahman, The associated Askey-Wilson polynomials, Trans. Amer. Math. Soc. 328(1991), 201-237.
20. M. E. H. Ismail and D. Stanton, On the Askey-Wilson and Rogers polynomials, Canad. J. Math. 40(1988), 1025-1045.
21. M. E. H. Ismail and J. Wilson, Asymptotic and generating relations for the q-Jacobi and the 4φ3 polynomials, J. Approx. Theory 36(1982), 43-54.
22. R. Koekoek and R. F. Swarttouw, The Askey scheme of hypergeometric orthogonal polynomials and its q-analogues, Report 94-05, Delft Univ. of Technology, 1994.
23. F. W. J. Olver, Asymptotics and Special Functions, Academic Press, New York, 1974.
24. P. I. Pastro, Orthogonal polynomials and some q-beta integrals of Ramanujan, J. Math. Anal. Appl. 112 (1985), 517-540.
25. 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.
26. J. Shohat and J. D. Tamarkin, The Problem of Moments, revised edition, Amer. Math. Soc., Providence, 1950.
27. G. Szegö, Beitrag zur Theorie der Thetafunktionen, Sitz. Preuss. Akad. Wiss. Phys. Math. Kl. XIX(1926), 242-252; reprinted In: Collected Papers, (ed. R. Askey), I, Birkhauser, Boston, 1982.
28. J. Wimp, Associated Jacobi polynomials and some applications, Canad. J. Math. 39(1987), 983-1000.