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q-Integral and Moment Representations for q-Orthogonal Polynomials

Published online by Cambridge University Press:  20 November 2018

Mourad E. H. Ismail  and
Dennis Stanton
Mourad E. H. Ismail
Affiliation:
Department of Mathematics, University of South Florida, Tampa, Florida 33620-5700, U.S.A.
Dennis Stanton
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455 U.S.A.
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Abstract

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We develop a method for deriving integral representations of certain orthogonal polynomials as moments. These moment representations are applied to find linear and multilinear generating functions for $q$-orthogonal polynomials. As a byproduct we establish new transformation formulas for combinations of basic hypergeometric functions, including a new representation of the $q$-exponential function $\text{ }{{\varepsilon }_{q}}$.

Keywords

33D4533D2033C4530E05q-integralq-orthogonal polynomialsassociated polynomialsq-difference equationsgenerating functionsAl-Salam-Chihara polynomialscontinuous q-ultraspherical polynomials
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 54 , Issue 4 , 01 August 2002 , pp. 709 - 735
Copyright
Copyright © Canadian Mathematical Society 2002

References

[1] Al-Salam, W. and Carlitz, L., Some orthogonal q-polynomials. Math. Nachr. 30 (1965), 47–61.Google Scholar
[2] Al-Salam, W. A. and Chihara, T. S., Convolutions of orthogonal polynomials. SIAM J. Math. Anal. 7 (1976), 16–28.Google Scholar
[3] Andrews, G. E. and Askey, R., Classical orthogonal polynomials. In: Polynomes Orthogonaux et Applications, (eds., C. Breziniski et al.), Lecture Notes in Math. 1171, Springer-Verlag, Berlin, 1984, 36–63.Google Scholar
[4] Askey, R. and Ismail, M. E. H., A generalization of ultraspherical polynomials. In: Studies in Pure Mathematics, (ed., P. Erdős), Birkhauser, Basel, 1983, 55–78.Google Scholar
[5] Askey, R. and Ismail, M. E. H., Recurrence relations, continued fractions and orthogonal polynomials. Mem. Amer. Math. Soc. 300(1984).Google Scholar
[6] Askey, R. A. and Wilson, J. A., Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc. 319(1985).Google Scholar
[7] Bustoz, J. and Ismail, M. E. H., The associated ultraspherical polynomials and their q-analogues. Canad. J. Math. 34 (1982), 718–736.Google Scholar
[8] Bustoz, J. and Suslov, S. K., Basic analog of Fourier series on a q-quadratic grid. Methods Appl. Anal. 5 (1998), 1–38.Google Scholar
[9] Charris, J. and Ismail, M. E. H., On sieved orthogonal polynomials, V: Sieved Pollaczek polynomials. SIAM J. Math. Anal. 18 (1987), 1177–1218.Google Scholar
[10] Gasper, G. and Rahman, M., Basic Hypergeometric Series. Cambridge University Press, Cambridge, 1990.Google Scholar
[11] Ismail, M. E. H., Rahman, M. and Suslov, S. K., Some summation theorems and transformations for q-series. Canad. J. Math. 49 (1997), 543–567.Google Scholar
[12] Ismail, M. E. H. and Stanton, D., On the Askey-Wilson and Rogers polynomials. Canad. J. Math. 40 (1988), 1025–1045.Google Scholar
[13] Ismail, M. E. H. and Stanton, D., Classical orthogonal polynomials as moments. Canad. J. Math. 49 (1997), 520–542.Google Scholar
[14] Ismail, M. E. H. and Stanton, D., More orthogonal polynomials as moments. Mathematical Essays in Honor of Gian-Carlo Rota, (eds., B. Sagan and R. P. Stanley), Birkhauser, Basel, 1998, 377–396.Google Scholar
[15] Ismail, M. E. H. and Zhang, R., Diagonalization of certain integral operators. Adv. Math. 109 (1994), 1–33.Google Scholar
[16] Koekoek, R. and Swarttouw, R., The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogues. Reports of the Faculty of Technical Mathematics and Information 94-05, Delft University of Technology, Delft, 1999.Google Scholar
[17] Milne-Thomson, L. M., The Calculus of Finite Differences. Macmillan, New York, 1933.Google Scholar
[18] Rahman, M., The associated classical orthogonal polynomials. In: Special Functions 2000, (eds., M. E. H. Ismail and S. Suslov), Kluwer, 2001.Google Scholar
[19] Rahman, M. and Tariq, Q., Poisson kernel for associated q-ultraspherical polynomials. Methods Appl. Anal. 4 (1997), 77–90.Google Scholar
[20] Shohat, J. and Tamarkin, J. D., The Problem of Moments. revised edition, Amer. Math. Soc., Providence, 1950.Google Scholar
[21] Szegő, G., Orthogonal Polynomials. Fourth Edition, Amer. Math. Soc., Providence, 1975.Google Scholar
[22] Tricomi, F. G., Integral equations Dover. New York, 1985.Google Scholar