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q-Hermite Polynomials and Classical Orthogonal Polynomials

  • Christian Berg (a1) and Mourad E. H. Ismail (a2)

Abstract

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.

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References

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1. Akhiezer, N.I., The Classical Moment Problem and Some Related Questions in Analysis. English translation, Oliver and Boyd, Edinburgh, 1965.
2. Al-Salam, W.A. and Carlitz, L., Some orthogonal q-polynomials, Math. Nachr. 30(1965), 47—61.
3. Al-Salam, W.A. and Chihara, T.S., Convolutions of orthogonal polynomials, SIAM J. Math. Anal. 7(1976), 1628.
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), 553561.
5. Al-Salam, W.A. and Verma, A., Q-analogs of some biorthogonal functions, Canad. Math. Bull. 26(1983), 225227.
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. 3663.
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. 151158.
8. Askey, R.A. and Ismail, M.E.H., A generalization of ultrasphericalpolynomials. In: Studies in Pure Math., (ed. Erdȍs, P.), Birkhauser, Basel, 1983. 5578.
9. Askey, R.A., Recurrence relations, continued fractions and orthogonal polynomials, Mem. Amer. Math. Soc. 300(1984).
10. Askey, R.A. and Wilson, J.A., Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 319(1985).
1. Askey, R.A., Associated Laguerre and Hermite polynomials, Proc. Royal Soc. Edinburgh Sect. A 96(1984), 1537.
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), 169209.
13. Bustoz, J. and Ismail, M.E.H., The associated ultraspherical polynomials and their q-analogues, Canad. J. Math. 34(1982), 718736.
14. Chihara, T.S., An Introduction to Orthogonal Polynomials. Gordon and Breach, New York, 1978.
15. Chihara, T.S. and Ismail, M.E.H., Extremal measures for a system of orthogonal polynomials, Constr. Approx. 9(1993), 111119.
16. Gasper, G. and Rahman, M., Basic Hypergeometric Series. Cambridge Univ. Press, Cambridge, 1990.
17. Ismail, M.E.H., A queueing model and a set of orthogonal polynomials, J. Math. Anal. Appl. 108(1985), 575594.
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), 61116.
19. Ismail, M.E.H. and Rahman, M., The associated Askey-Wilson polynomials, Trans. Amer. Math. Soc. 328(1991), 201237.
20. Ismail, M.E.H. and Stanton, D., On the Askey-Wilson and Rogers polynomials, Canad. J. Math. 40(1988), 10251045.
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), 4354.
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.
23. Olver, F.W.J., Asymptotics and Special Functions. Academic Press, New York, 1974.
24. Pastro, P.I., Orthogonal polynomials and some q-beta integrals of Ramanujan, J. Math. Anal. Appl. 112 (1985), 517540.
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), 14211431.
26. Shohat, J. and Tamarkin, J.D., The Problem of Moments. revised edition, Amer. Math. Soc, Providence, 1950.
27. Szegö, G., Beitragzur Theorie der Thetafunktionen. Sitz. Preuss. Akad. Wiss. Phys. Math. Kl. XIX(1926), 242252. reprinted In: Collected Papers, (ed. Askey, R.), I, Birkhauser, Boston, 1982.
28. Wimp, J., Associated Jacobi polynomials and some applications, Canad. J. Math. 39(1987), 983—1000.
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q-Hermite Polynomials and Classical Orthogonal Polynomials

  • Christian Berg (a1) and Mourad E. H. Ismail (a2)

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