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On q-Exponential Functions for |q| = 1

Published online by Cambridge University Press:  20 November 2018

D. S. Lubinsky
D. S. Lubinsky*
Affiliation:
Department of Mathematics Witwatersrand University Wits 2050 South Africa, e-mail: 036dsl@cosmos.wits.ac.za
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Abstract

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We discuss the $q$-exponential functions ${{e}_{q}},\,{{E}_{q}}$ for $q$ on the unit circle, especially their continuity in $q$, and analogues of the limit relation ${{\lim }_{q\to 1}}\,{{e}_{q}}\,((1-q)z)\,=\,{{e}^{z}}.$

Keywords

33D0511A5511K70q-seriesq-exponentials
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 41 , Issue 1 , 01 March 1998 , pp. 86 - 97
Copyright
Copyright © Canadian Mathematical Society 1998

References

1. Askey, R., The q-gamma and q-beta functions. Appl. Anal. 8 (1978), 125–141.Google Scholar
2. Askey, R., Ramanujan's Extensions of the Gamma and Beta Functions. Amer. Math. Monthly 87 (1980), 346–359.Google Scholar
3. Driver, K. A., Convergence of Padé Approximants for Some q-hypergeometric Series (Wynn's Power Series I,II and III). Ph. D Thesis, Witwatersrand University, Johannesburg, 1991.Google Scholar
4. Driver, K. A. and Lubinsky, D. S., Convergence of Padé Approximants for a q-hypergeometric Series (Wynn's Power Series I). Aequationes Math. 42 (1991), 85–106.Google Scholar
5. Driver, K. A. and Lubinsky, D. S., Convergence of Padé Approximants for a q-hypergeometric Series (Wynn's Power Series II). Colloq. Math. (Janos Bolyai Society) 58 (1990), 221–239.Google Scholar
6. Driver, K. A. and Lubinsky, D. S., Convergence of Padé Approximants for a q-hypergeometric Series (Wynn's Power Series III). Aequationes Math. 45 (1993), 1–23.Google Scholar
7. Driver, K. A., Lubinsky, D. S., Petruska, G. and Sarnak, P., Irregular Distribution of fnågÒ n = 1Ò2Ò3Ò ð ð ð. Quadrature of Singular Integrands and Curious Basic Hypergeometric Series, Indag. Math. 2 (1991), 469–481.Google Scholar
8. Exton, R., q-Hypergeometric Functions and Applications. Ellis Horwood, Chichester, 1983.Google Scholar
9. Fine, N. J., Basic Hypergeometric Series and Applications. Math. Surveys and Monographs 27, Amer. Math. Soc., Providence, 1988.Google Scholar
10. Gasper, G. and Rahman, M., Basic Hypergeometric Series. Cambridge University Press, Cambridge, 1990.Google Scholar
11. Hardy, G. H. and Littlewood, J. E., Note on the Theory of Series (XXIV): A curious Power Series. Proc. Cambridge Philos. Soc. 42 (1946), 85–90.Google Scholar
12. Hardy, G. H. and Wright, E. M., An Introduction to the Theory of Numbers. Oxford University Press, Oxford, 1975, Fourth Edition with corrections.Google Scholar
13. Hille, E. M., Analytic Function Theory. Vol. 2, Chelsea, New York, 1987.Google Scholar
14. Lubinsky, D. S., Note on Polynomial Approximation of Monomials and Diophantine Approximation. J. Approx. Theory 43 (1985), 29–35.Google Scholar
15. Lubinsky, D. S. and Saff, E. B., Convergence of Padé Approximants of Partial Theta Functions and the Rogers-Szegö Polynomials. Constr. Approx. 3 (1987), 331–361.Google Scholar
16. McIntosh, R. J., Some Asymptotic Formulae for Ramanujan's Mock-Theta Functions. In: Contemporary Mathematics 166, (eds. G. Andrews et al), Amer. Math Soc., Providence, 1994, 189–196.Google Scholar
17. Pastro, P. I., Orthogonal Polynomials and Some q-beta integrals of Ramanujan. J. Math. Anal. Appl. 112 (1985), 517–540.Google Scholar
18. Petruska, G., On the Radius of Convergence of q-Series. Indag. Math. 3 (1992), 353–364.Google Scholar
19. Spiridonov, V. and Zhedanov, A., Discrete Darboux Transformations, Discrete time Toda lattice and the Askey-Wilson polynomials. Methods Appl. of Anal. 2 (1995), 369–398.Google Scholar
20. Spiridonov, V. and Zhedanov, A., Zeros and Orthogonality of the Askey-Wilson Polynomials for q a Root of Unity. manuscript.Google Scholar
21. Zhedanov, A., On the Polynomials Orthogonal on Regular Polygons. manuscript.Google Scholar