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Quadratic $q$-exponentials
and connection coefficient problems


Authors: Mourad E. H. Ismail, Mizan Rahman and Dennis Stanton
Journal: Proc. Amer. Math. Soc. 127 (1999), 2931-2941
MSC (1991): Primary 33D45; Secondary 42C15
DOI: https://doi.org/10.1090/S0002-9939-99-05017-0
Published electronically: April 23, 1999
MathSciNet review: 1621949
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Abstract | References | Similar Articles | Additional Information

Abstract: We establish expansion formulas of $q$-exponential functions in terms of continuous $q$-ultraspherical polynomials, continuous $q$-Hermite polynomials and Askey-Wilson polynomials. The proofs are based on solving connection coefficient problems.


References [Enhancements On Off] (What's this?)

  • [Al:Ch] W. A. Al-Salam and T. S. Chihara, Convolutions of orthogonal polynomials, SIAM J. Math. Anal. 7 (1976), 16-28. MR 53:3381
  • [An] G. E. Andrews, Connection coefficient problems and partitions, Proc. Symp. Pure Math 34 (1979), 1-24. MR 80c:33004
  • [As:Is] R. A. Askey and M. E. H. Ismail, A generalization of ultraspherical polynomials, Studies in Pure Mathematics (P. Erdös, ed.), Birkhauser, Basel, 1983, pp. 55-78. MR 87a:33015
  • [As:Is2] R. A. Askey and M. E. H. Ismail, Recurrence relations, continued fractions and orthogonal polynomials, Memoirs Amer. Math. Soc. 49 (1984). MR 85g:33008
  • [As:Wi] R. A. Askey and J. A. Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Memoirs Amer. Math. Soc. 54 (1985). MR 87a:05023
  • [At:Su] N. M. Atakishiev and S. K. Suslov, Difference hypergeometric functions, Progress in Approximation Theory, An International Perspective (A. A. Gonchar and E. B. Saff, ed.), Springer-Verlag, New York, 1992, pp. 1-35. MR 94k:33039
  • [Fi:Is] J. Fields and M. E. H. Ismail, Polynomial expansions, Mathematics of Computation 29 (1975), 894-902. MR 51:8680
  • [Fl:Vi] R. Floreanini and L. Vinet, A model for the continuous $q$-ultraspherical polynomials, J. Math. Phys. 36 (1995), 3800-3813. MR 96d:33012
  • [Fl:Le:Vi] R. Floreanini, J. LeTourneux and L. Vinet, More on the q-oscillator algebra and q-orthogonal polynomials, Journal of Physics A 28 (1995), L287-L293. MR 96e:33043
  • [Fl:Le:Vi2] R. Floreanini, J. LeTourneux and L. Vinet, Symmetry techniques for the Al-Salam-Chihara polynomials, J. Phys. A 30 (1997), 3107-3114. MR 98k:33036
  • [Ga:Ra] G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge University Press, Cambridge, 1990. MR 91d:33034
  • [Ge:St] I. Gessel and D. Stanton, Applications of $q$-Lagrange inversion to basic hypergeometric series, Trans. Amer. Math. Soc. 277 (1983), 173-201. MR 84f:33009
  • [Is] M. E. H. Ismail, The zeros of basic Bessel functions the functions $J_{\nu +ax}\left (x\right ) $ and the associated orthogonal polynomials, J. Math. Anal. Appl. 86 (1982), 1-19. MR 83c:33010
  • [Is:Ma:Su] M. E. H. Ismail, D. Masson and S. Suslov, The $q$-Bessel functions on $q$-quadratic grid (to appear).
  • [Is:Ra:Zh] M. E. H. Ismail, M. Rahman and R. Zhang, Diagonalization of certain integral operators II, J. Comp. Appl. Math. 68 (1996), 163-196. MR 98d:33011
  • [Is:Zh] M. E. H. Ismail and R. Zhang, Diagonalization of certain integral operators, Advances in Math. 109 (1994), 1-33. MR 96d:39005
  • [Lu] Y. Luke, The Special Functions and Their Approximations, volume 2, Academic Press, New York, 1969. MR 40:2909
  • [Ra] M. Rahman, An integral representation and some transformation properties of $q$-Bessel functions, J. Math. Anal. Appl. 125 (1987), 58-71. MR 88h:33020
  • [Su] S. K. Suslov, Addition theorems for some $q$-exponential and trigonometric functions, Methods and Applications of Analysis 4 (1997), 11-32. CMP 97:14

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Additional Information

Mourad E. H. Ismail
Affiliation: Department of Mathematics, University of South Florida, Tampa, Florida 33620-5700
Email: ismail@math.nsf.edu

Mizan Rahman
Affiliation: Department of Mathematics, Carleton University, Ottawa, Ontario, Canada K1S 5B6
Email: mrahman@math.carleton.ca

Dennis Stanton
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Email: stanton@math.umn.edu

DOI: https://doi.org/10.1090/S0002-9939-99-05017-0
Keywords: Askey-Wilson polynomials, continuous $q$-ultraspherical polynomials, $q$-exponential functions, $q$-Bessel functions
Received by editor(s): December 29, 1997
Published electronically: April 23, 1999
Additional Notes: The first author was partially supported by NSF grant DMS-9625459, the second author was partially supported by NSERC grant A6197, and the third author was partially supported by NSF grant DMS-9400510.
Communicated by: Hal. L. Smith
Article copyright: © Copyright 1999 American Mathematical Society

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