Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-24T17:06:52.079Z Has data issue: false hasContentIssue false
  • English
  • Français

Some Summation Theorems and Transformations for Q-Series

Published online by Cambridge University Press:  20 November 2018

Mourad E. H. Ismail ,
Mizan Rahman  and
Sergei K. Suslov
Mourad E. H. Ismail
Affiliation:
Department of Mathematics, University of South Florida, Tampa, Florida 33620, USA
Mizan Rahman
Affiliation:
Department of Mathematics and Statistics, Carleton University, Ottawa, Ontario, K1S 5B6
Sergei K. Suslov
Affiliation:
Department of Mathematics, Arizona State University, Tempe, Arizona 85287, USA
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We introduce a double sum extension of a very well-poised series and extend to this the transformations of Bailey and Sears as well as the 6ϕ5 summation formula of F. H. Jackson and the q-Dixon sum. We also give q-integral representations of the double sum. Generalizations of the Nassrallah-Rahman integral are also found.

Keywords

33D2033D60Basic hypergeometric seriesbalanced seriesvery well-poised seriesintegral representationsAl-Salam-Chihara polynomials
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 49 , Issue 3 , 01 June 1997 , pp. 543 - 567
Copyright
Copyright © Canadian Mathematical Society 1997

References

1. Al-Salam, W. A. and Chihara, T.S., Convolutions of orthogonal polynomials, SIAM J. Math. Anal. 7(1976), 16–28.Google Scholar
2. Askey, R.A. and Ismail, M. E. H., Recurrence relations, continued fractions and orthogonal polynomials, Mem. Amer. Math. Soc. 300(1984).Google Scholar
3. Askey, R.A., M. Rahman, and Suslov, S.K., On a general q-Fourier transformation with nonsymmetric kernels, J. Comput. Appl. Math. 68(1996), 25–55.Google Scholar
4. Askey, R.A. and Wilson, J.A., Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer.Math. Soc. 319(1985).Google Scholar
5. Berg, C. and Ismail, M. E. H., q-Hermite polynomials and the classical polynomials, Canad. J. Math. 48(1996), 43–63.Google Scholar
6. Gasper, G. and Rahman, M., Basic Hypergeometric Series, Cambridge Univ. Press, Cambridge, 1990.Google Scholar
7. Ismail, M. E. H., Poisson kernels, in preparation.Google Scholar
8. Ismail, M. E. H. and D. Stanton, Classical orthogonal polynomials as moments, Canad. J. Math., to appear.Google Scholar
9. Ismail, M. E. H. and Wilson, J.A., Asymptotic and generating relations for q-Jacobi and 4û3 polynomials, J. Approx. Theory 36(1982), 43–54.Google Scholar
10. Nassrallah, B. and Rahman, M., Projection formulas, a reproducing kernel and generating functions for q-Wilson polynomials, SIAM J. Math. Anal. 16(1985), 186–197.Google Scholar
11. Suslov, S.K., unpublished notes.Google Scholar