On the Completeness of Sets of q-Bessel Functions Jν(3)(x; q)

  • L. D. Abreu
  • J. Bustoz
Part of the Developments in Mathematics book series (DEVM, volume 13)


We study completeness of systems of third Jackson q-Bessel functions by two quite different methods. The first uses a Dalzell-type criterion and relies on orthogonality and the evaluation of certain q-integrals. The second uses classical entire function theory.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Abreu, L. D., Bustoz, J., and Cardoso, J. L. (2003). The roots of the third Jackson q-Bessel function. Internat. J. Math. Math. Sci., 67:4241–4248.CrossRefMathSciNetGoogle Scholar
  2. Boas, R. P. and Pollard, H. (1947). Complete sets of Bessel and Legendre functions. Ann. of Math., 48:366–384.CrossRefMathSciNetGoogle Scholar
  3. Dalzell, D. P. (1945). On the completeness of a series of normal orthogonal functions. J. Lond. Math. Soc., 20:87–93.zbMATHMathSciNetGoogle Scholar
  4. Exton, H. (1983). q-Hypergeometric Functions and Applications. Ellis Horwood, Chichester.Google Scholar
  5. Higgins, J. R. (1977). Completeness and basis properties of sets of special functions. Cambridge University Press, London, New York, Melbourne.Google Scholar
  6. Ismail, M. E. H. (1982). The zeros of basic Bessel functions, the functions Jv+ax(x), and associated orthogonal polynomials. J. Math. Anal. Appl., 86:1–19.CrossRefzbMATHMathSciNetGoogle Scholar
  7. Ismail, M. E. H. (2003). Some properties of jackson's third q-bessel function. Preprint.Google Scholar
  8. Jackson, F. H. (1904). On generalized functions of Legendre and Bessel. Transactions of the Royal Society of Edinburgh, 41:1–28.zbMATHGoogle Scholar
  9. Koelink, H. T. (1999). Some basic Lommel polynomials. Journal of Approximation Theory, 96:345–365.CrossRefzbMATHMathSciNetGoogle Scholar
  10. Koelink, H. T. and Swarttouw, R. F. (1994). On the zeros of the Hahn-Exton q-Bessel function and associated q-Lommel polynomials. J. Math. Anal. Appl., 186:690–710.CrossRefMathSciNetGoogle Scholar
  11. Kvitsinsky, A. A. (1995). Spectral zeta functions for q-Bessel equations. J. Phys. A: Math. Gen., 28:1753–1764.CrossRefzbMATHMathSciNetGoogle Scholar
  12. Levin, B. Y. (1980). Distribution of zeros of Entire Functions. American Mathematical Society, Providence, RI.Google Scholar
  13. Swartouw, R. F. (1992). The Hahn-Exton q-Bessel Function. PhD thesis, Technische Universiteit Delft.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • L. D. Abreu
    • 1
  • J. Bustoz
    • 2
  1. 1.Department of MathematicsUniversidade de CoimbraCoimbraPortugal
  2. 2.Department of MathematicsArizona State UniversityTempe

Personalised recommendations