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On the Askey-Wilson and Rogers Polynomials

Published online by Cambridge University Press:  20 November 2018

Mourad E. H. Ismail  and
Dennis Stanton
Mourad E. H. Ismail
Affiliation:
Arizona State University, Tempe, Arizona
Dennis Stanton
Affiliation:
University of Minnesota, Minneapolis, Minnesota
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The q-shifted factorial (a)n or (a; q)n is

and an empty product is interpreted as 1. Recently, Askey and Wilson [6] introduced the polynomials

1.1

where

1.2

and

1.3

We shall refer to these polynomials as the Askey-Wilson polynomials or the orthogonal 4ϕ3 polynomials. They generalize the 6 — j symbols and are the most general classical orthogonal polynomials, [2].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 40 , Issue 5 , 01 October 1988 , pp. 1025 - 1045
Copyright
Copyright © Canadian Mathematical Society 1988

References

1. Al-Salam, W. and Carlitz, L., Some orthogonal q-polynomials, Math. Nachr. 30 (1965), 47–61.Google Scholar
2. Andrews, G. and Askey, R. A., Classical orthogonal polynomials, to appear.Google Scholar
3. Askey, R. A., An elementary evaluation of a beta type integral, Indian J. Pure Appl. Math. 74(1983), 892–895.Google Scholar
4. Askey, R. A. and Ismail, M. E. H., The Rogers q-ultraspherical polynomials, Approximation Theory III (Academic Press, New York, 1980), 175–182.Google Scholar
5. Askey, R. A. and Ismail, M. E. H., A generalization of ultraspherical polynomials, Studies in Pure Mathematics (Birkhauser, Basel, 1983).CrossRefGoogle Scholar
6. Askey, R. A. and Wilson, J. A., Some basic hyper geometric orthogonal polynomials that generalize Jacobi polynomials, Memoirs Amer. Math. Soc. 319 (1985), to appear.Google Scholar
7. Bressoud, D., Linearization and related formulas for q-ultraspherical polynomials, SIAM J. Math. Anal. 12 (1981), 161–168.Google Scholar
8. Carlitz, L., Some extensions of the Mehler formula, Collectanea Mathematica 21 (1970), 117–130.Google Scholar
9. Chihara, T., An introduction to orthogonal polynomials (Gordon and Breach, New York, 1978).Google Scholar
10. Foata, D., Some Hermite polynomial identities and their combinatorics, Advances Appl. Math. 2 (1981), 250–259.Google Scholar
11. Ismail, M. E. H., A queueing model and a set of orthogonal polynomials, J. Math. Anal. Appl. 130 (1985), to appear.Google Scholar
12. Ismail, M. E. H., Stanton, D. and Viennot, G., The combinatorics of the q-Hermite polynomials and the Askey-Wilson integral, Eur. J. Comb. 8 (1987), 379–392.Google Scholar
13. Kibble, W. F., An extension of theorem of Mehler on Hermite polynomials, Proc. Cambridge Philos. Soc. 41 (1945), 12–15.Google Scholar
14. Louck, J. D., Extension of the Kibble-Slepian formula for Hermite polynomials using Boson operator methods, Advances in Appl. Math. 2 (1981), 239–249.Google Scholar
15. Nassrallah, B. and Rahman, M., Projection formulas, a reproducing kernel and a generating function for q-Wilson polynomials, SIAM J. Math. Anal. 16 (1985), 186–197.Google Scholar
16. Rahman, M., A simple evaluation of Askey and Wilson's q-integral, Proc. Amer. Math. Soc. 92 (1984), 413–417.Google Scholar
17. Rainville, E. D., Special functions (Chelsea Publishing Co., Bronx, New York, 1971).Google Scholar
18. Rogers, L. J., On the expansion of some infinite products, Proc. London Math. Soc. 24 (1893), 337–352.Google Scholar
19. Rogers, L. J., Second memoir on the expansion of certain infinite products, Proc. London Math. Soc. 25 (1894), 318–343.Google Scholar
20. Rogers, L. J., Third memoir on the expansion of certain infinite products, Proc. London Math. Soc. 26 (1895), 15–32.Google Scholar
21. Sears, D., Transformations of basic hyper geometric functions of any order, Proc. London Math. Soc. 55(1951), 181–191.Google Scholar
22. Slepian, D., On the symmetrized Kronecker power of a matrix and extensions of Mehler's formula for H ermite polynomials, SIAM J. Math. Anal. 3 (1972), 606–616.Google Scholar
23. Szegö, G., Ein Beitrag zur théorie der Thetafunktionen, Sitz. Preuss. Akad. Wiss. Phys. Math. KL, XIX (1926), 242–252, Collected Papers (Birkhauser, Boston, 1982), 795–805.Google Scholar
24. Szegö, G., Orthogonal polynomials, Fourth Edition, Amer. Math. Soc. Colloquium Publications 23 (Amer. Math. Soc, Providence, R.I., 1975).Google Scholar
25. Tricomi, F. G., Integral equations (Interscience, New York, 1957).Google Scholar