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Complex Weight Functions for Classical Orthogonal Polynomials

Published online by Cambridge University Press:  20 November 2018

Mourad E. H. Ismail ,
David R. Masson  and
Mizan Rahman
Mourad E. H. Ismail
Affiliation:
Department of Mathematics, University of South Florida, Tampa, Florida 33620, U. S. A.
David R. Masson
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario M5S 1A1
Mizan Rahman
Affiliation:
Department of Mathematics, Carleton University, Ottawa, Ontario K1S 1B6
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Abstract

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We give complex weight functions with respect to which the Jacobi, Laguerre, little q-Jacobi and Askey-Wilson polynomials are orthogonal. The complex functions obtained are weight functions in a wider range of parameters than the real weight functions. They also provide an alternative to the recent distributional weight functions of Morton and Krall, and the more recent hyperfunction weight functions of Kim.

Keywords

33A6533A3042C05Complex weight functionsreal weight functionsJacobiLaguerrelittle q-JacobiAskey-Wilson polynomialsdistributional weight functionshyperfunction weight functionsthree term recurrence relationsinner products.
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 43 , Issue 6 , 01 December 1991 , pp. 1294 - 1308
Copyright
Copyright © Canadian Mathematical Society 1991

References

1. Andrews, G. and Askey, R., Classical orthogonal polynomials. In Polynômes Orthogonaux et Applications, Lecture Notes in Mathematics 1171, ed. C. Brezinski et al., Springer-Verlag, New York, 1985, 36–62.Google Scholar
2. Askey, R. and Ismail, M.E.H., A generalization of the ultraspherical polynomials. In: Studies in Pure Mathematics, éd. P. Erdős, Birkhauser-Verlag, Basel, 1983, 55–78.Google Scholar
3. Askey, R. and Wilson, J.A., A set of orthogonal polynomials that generalize Jacobi polynomials. Memoirs Amer. Math. Soc. 319, 1985.Google Scholar
4. Atakishiyev, N.M. and Suslov, S.K., On the Askey-Wilson polynomials, Constructive Approximation (1992), to appear.Google Scholar
5. Chihara, T.S., An introduction to orthogonal polynomials. Gordon and Breach, New York, 1978.Google Scholar
6. Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F.G., Higher transcendental functions, Volume 1. McGraw-Hill, New York, 1953.Google Scholar
7. Erdelyi, A., Higher transcendental functions, Volume 2. McGraw-Hill, New York, 1953.Google Scholar
8. Fine, N., Basic hypergeometric series and applications, Mathematical surveys and monographs. American Mathematical Society, Providence, Rhode Island, 27, 1988.Google Scholar
9. Gasper, G. and Rahman, M., Basic hypergeometric series. Cambridge University Press, Cambridge, 1990.Google Scholar
10. Gupta, D.P. and Masson, D., Exceptional q-Askey-Wilson polynomials and continuedfractions, Proc. Amer. Math. Soc, 112(1991), 717–727.Google Scholar
11. Kim, K., Hyperfunctions and orthogonal polynomials, to appear.Google Scholar
12. Masson, D.R., Wilson polynomials and some continued fractions ofRamanujan, Rocky Mountain J. Math. 21(1991),489–499.Google Scholar
13. Morton, R. and Krall, A., Distributional weight functions for orthogonal polynomials, SIAM J. Math. Anal. 9(1978), 604–626.Google Scholar
14. Rahman, M., q-Wilson functions of the secondkind, SIAM J. Math. Anal. 17(1986), 1280–1286.Google Scholar
15. Rainville, E.D., Special functions. Reprinted by Chelsea, Bronx, New York, 1971.Google Scholar
16. Rusev, P., Analytic functions and classical orthogonal polynomials. Publishing House of the Bulgarian Academy of Sciences, Sofia, 1984.Google Scholar
17. Szegö, G., Orthogonal polynomials. Colloquium Publications, fourth edition, 23, American Mathematical Society, Providence, Rhode Island, 1975.Google Scholar