Advertisement

New Proofs of Some q-Series Results

  • Mourad E.H. Ismail
  • Ruiming Zhang
Chapter
Part of the Developments in Mathematics book series (DEVM, volume 13)

Abstract

We use a Sheffer classification technique to give very short proofs of the addition theorem for the ε q function, the representation of ε q as a multiple of a 2ϕ1, and a relatively new representation of ε q . A direct proof of the evaluation of the connection coefficients of the Askey-Wilson polynomials and the Nassrallah-Rahman integral are also given. A sim ple proof of a characterization theorem for the continuous q-Hermite polynomials is also given.

Keywords

Sheffer classification delta operators polynomial bases Askey-Wilson operators the q-exponential function εq 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Al-Salam, W. A. (1995). Characterization of the Rogers q-Hermite polynomials. Internat. J. Math. Math. Sci., 18:641–647.CrossRefzbMATHMathSciNetGoogle Scholar
  2. Andrews, G. E., Askey, R. A., and Roy, R. (1999). Special Functions. Cambridge University Press, Cambridge.zbMATHGoogle Scholar
  3. Askey, R. A. (1989). Continuous q-Hermite polynomials when q > 1. In Stanton, D., editor, q-Series and Partitions, IMA Volumes in Mathematics and Its Applications, pages 151–158. Springer-Verlag, New York.Google Scholar
  4. Askey, R. A. and Ismail, M. E. H. (1983). A generalization of ultraspherical polynomials. In Erdős, P., editor, Studies in Pure Mathematics, pages 55–78. Birkhauser, Basel.Google Scholar
  5. Askey, R. A. and Wilson, J. A. (1985). Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Memoirs Amer. Math. Soc., 54(319):iv+55.MathSciNetGoogle Scholar
  6. Brown, B. M., Evans, W. D., and Ismail, M. E. H. (1996). The Askey-Wilson polynomials and q-Sturm-Liouville problems. Math. Proc. Cambridge Phil. Soc., 119:1–16.MathSciNetCrossRefzbMATHGoogle Scholar
  7. Cooper, S. (1996). The Askey-Wilson operator and the 6ϕ5 summation formula. Preprint.Google Scholar
  8. Gasper, G. and Rahman, M. (1990). Basic Hypergeometric Series. Cambridge University Press, Cambridge.zbMATHGoogle Scholar
  9. Ismail, M. E. H. (1995). The Askey-Wilson operator and summation theorems. In Ismail, M., Nashed, M. Z., Zayed, A., and Ghaleb, A., editors, Mathematical Analysis, Wavelets, and Signal Processing, volume 190 of Contemporary Mathematics, pages 171–178. American Mathematical Society, Providence.Google Scholar
  10. Ismail, M. E. H. (2001). An operator calculus for the Askey-Wilson operator. Annals of Combinatorics, 5:333–348.CrossRefMathSciNetGoogle Scholar
  11. Ismail, M. E. H. and Stanton, D. (2003a). Applications of q-Taylor theorems. J. Comp. Appl. Math., 153:259–272.CrossRefMathSciNetzbMATHGoogle Scholar
  12. Ismail, M. E. H. and Stanton, D. (2003b). q-Taylor theorems, polynomial expansions, and interpolation of entire functions. J. Approx. Theory, 123:125–146.CrossRefMathSciNetzbMATHGoogle Scholar
  13. Ismail, M. E. H. and Zhang, R. (1994). Diagonalization of certain integral operators. Advances in Math., 109:1–33.CrossRefMathSciNetzbMATHGoogle Scholar
  14. Nassrallah, B. and Rahman, M. (1985). Projection formulas, a reproducing kernel and a generating function for q-Wilson polynomials. SIAM J. Math. Anal., 16:186–197.CrossRefMathSciNetzbMATHGoogle Scholar
  15. Rainville, E. D. (1971). Special Functions. Chelsea Publishing Co., Bronx, NY.zbMATHGoogle Scholar
  16. Rota, G.-C., Kahaner, D., and Odlyzko, A. (1973). On the foundations of combinatorial theory. VIII. Finite operator calculus. J. Math. Anal. Appl., 42:684–760.CrossRefMathSciNetzbMATHGoogle Scholar
  17. Suslov, S. K. (1997). Addition theorems for some q-exponential and trigonometric functions. Methods and Applications of Analysis, 4:11–32.zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • Mourad E.H. Ismail
    • 1
  • Ruiming Zhang
    • 2
  1. 1.Department of MathematicsUniversity of Central FloridaOrlando
  2. 2.Department of MathematicsUniversity of South FloridaTampa

Personalised recommendations