Quadratic -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 -exponential functions in terms of continuous -ultraspherical polynomials, continuous -Hermite polynomials and Askey-Wilson polynomials. The proofs are based on solving connection coefficient problems.

**[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 -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 -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 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 -Bessel functions on -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 -Bessel functions*, J. Math. Anal. Appl.**125**(1987), 58-71. MR**88h:33020****[Su]**S. K. Suslov,*Addition theorems for some -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