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q-Hermite Polynomials and Classical Orthogonal Polynomials

Published online by Cambridge University Press:  20 November 2018

Christian Berg  and
Mourad E. H. Ismail
Christian Berg
Affiliation:
Mathematics Institute Copenhagen University Universitetsparken5 DK-2100 Copenhagen Ø Denmark, e-mail: [email protected]
Mourad E. H. Ismail
Affiliation:
Department of Mathematics University of South Florida Tampa, Florida 33620-5700 USA., e-mail: [email protected]
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Abstract

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We use generating functions to express orthogonality relations in the form of q-beta. integrals. The integrand of such a q-beta. integral is then used as a weight function for a new set of orthogonal or biorthogonal functions. This method is applied to the continuous q-Hermite polynomials, the Al-Salam-Carlitz polynomials, and the polynomials of Szegö and leads naturally to the Al-Salam-Chihara polynomials then to the Askey-Wilson polynomials, the big q-Jacobi polynomials and the biorthogonal rational functions of Al-Salam and Verma, and some recent biorthogonal functions of Al-Salam and Ismail.

Keywords

33D4533A6544A60Askey-Wilson polynomialsq-orthogonal polynomialsorthogonality relationsq-beta integrals, q-Hermite polynomialsbiorthogonal rational functions
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 48 , Issue 1 , 01 February 1996 , pp. 43 - 63
Copyright
Copyright © Canadian Mathematical Society 1997

References

1. Akhiezer, N.I., The Classical Moment Problem and Some Related Questions in Analysis. English translation, Oliver and Boyd, Edinburgh, 1965.Google Scholar
2. Al-Salam, W.A. and Carlitz, L., Some orthogonal q-polynomials, Math. Nachr. 30(1965), 47—61.Google Scholar
3. Al-Salam, W.A. and Chihara, T.S., Convolutions of orthogonal polynomials, SIAM J. Math. Anal. 7(1976), 16–28.Google Scholar
4. Al-Salam, W.A. and Ismail, M.E.H., A q-beta integral on the unit circle and some biorthogonal rational functions, Proc. Amer. Math. Soc. 121(1994), 553–561.Google Scholar
5. Al-Salam, W.A. and Verma, A., Q-analogs of some biorthogonal functions, Canad. Math. Bull. 26(1983), 225–227.Google Scholar
6. Andrews, G.E. and Askey, R.A., Classical orthogonal polynomials. In: Polynomes Orthogonaux et Applications, (eds. Brezinski, C., et al), Lecture Notes in Math. 1171, Springer-Verlag, Berlin, 1984. 36–63.Google Scholar
7. Askey, R.A., Continuous q-Hermite polynomials when q <. 1. In: q-Series and Partitions, (ed. Stanton, D.), IMA Math. Appl., Springer-Verlag, New York, 1989. 151–158.Google Scholar
8. Askey, R.A. and Ismail, M.E.H., A generalization of ultrasphericalpolynomials. In: Studies in Pure Math., (ed. Erdȍs, P.), Birkhauser, Basel, 1983. 55–78.Google Scholar
9. Askey, R.A., Recurrence relations, continued fractions and orthogonal polynomials, Mem. Amer. Math. Soc. 300(1984).Google Scholar
10. Askey, R.A. and Wilson, J.A., Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 319(1985).Google Scholar
1. Askey, R.A., Associated Laguerre and Hermite polynomials, Proc. Royal Soc. Edinburgh Sect. A 96(1984), 15–37.Google Scholar
12. Berg, C. and Valent, G., The Nevanlinna parameterization for some indeterminate Stieltjes moment problems associated with birth and death processes, Methods Appl. Anal. 1(1994), 169–209.Google Scholar
13. Bustoz, J. and Ismail, M.E.H., The associated ultraspherical polynomials and their q-analogues, Canad. J. Math. 34(1982), 718–736.Google Scholar
14. Chihara, T.S., An Introduction to Orthogonal Polynomials. Gordon and Breach, New York, 1978.Google Scholar
15. Chihara, T.S. and Ismail, M.E.H., Extremal measures for a system of orthogonal polynomials, Constr. Approx. 9(1993), 111–119.Google Scholar
16. Gasper, G. and Rahman, M., Basic Hypergeometric Series. Cambridge Univ. Press, Cambridge, 1990.Google Scholar
17. Ismail, M.E.H., A queueing model and a set of orthogonal polynomials, J. Math. Anal. Appl. 108(1985), 575–594.Google Scholar
18. Ismail, M.E.H. and Masson, D.R., Q-Hermite polynomials, biorthogonal rational functions and Q-beta integrals, Trans. Amer. Math. Soc. 346(1994), 61–116.Google Scholar
19. Ismail, M.E.H. and Rahman, M., The associated Askey-Wilson polynomials, Trans. Amer. Math. Soc. 328(1991), 201–237.Google Scholar
20. Ismail, M.E.H. and Stanton, D., On the Askey-Wilson and Rogers polynomials, Canad. J. Math. 40(1988), 1025–1045.Google Scholar
21. Ismail, M.E.H. and Wilson, J., Asymptotic and generating relations for the q-Jacobi and the 4-3 polynomials, J. Approx. Theory 36(1982), 43–54.Google Scholar
22. Koekoek, R. and Swarttouw, R.F., The Askey scheme of hypergeometric orthogonal polynomials and its q-analogues, Repor. 94–05. Delft Univ. of Technology, 1994.Google Scholar
23. Olver, F.W.J., Asymptotics and Special Functions. Academic Press, New York, 1974.Google Scholar
24. Pastro, P.I., Orthogonal polynomials and some q-beta integrals of Ramanujan, J. Math. Anal. Appl. 112 (1985), 517–540.Google Scholar
25. Rahman, M., Biorthogonality of a system of rational functions with respect to a positive measure on [— 1,1 ], SIAM J. Math. Anal. 22(1991), 1421–1431.Google Scholar
26. Shohat, J. and Tamarkin, J.D., The Problem of Moments. revised edition, Amer. Math. Soc, Providence, 1950.Google Scholar
27. Szegö, G., Beitragzur Theorie der Thetafunktionen. Sitz. Preuss. Akad. Wiss. Phys. Math. Kl. XIX(1926), 242–252. reprinted In: Collected Papers, (ed. Askey, R.), I, Birkhauser, Boston, 1982.Google Scholar
28. Wimp, J., Associated Jacobi polynomials and some applications, Canad. J. Math. 39(1987), 983—1000.Google Scholar