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Functions $ q$-orthogonal with respect to their own zeros

Author: Luis Daniel Abreu
Journal: Proc. Amer. Math. Soc. 134 (2006), 2695-2701
MSC (2000): Primary 42C05, 33D45; Secondary 39A13
Published electronically: March 23, 2006
MathSciNet review: 2213749
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Abstract: In 1939, G. H. Hardy proved that, under certain conditions, the only functions satisfying

$\displaystyle \int_{0}^{1}f(\lambda _{m}t)f(\lambda _{n}t)dt=0, $

where the $ \lambda _{n}$ are the zeros of $ f$, are the Bessel functions. We replace the above integral by the Jackson $ q$-integral and give the $ q$-analogue of Hardy's result.

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

  • 1. L. D. Abreu, A q-Sampling Theorem related to the q-Hankel transform, Proc. Amer. Math. Soc. 133 (2005), 1197-1203. MR 2117222
  • 2. W. N. Everitt, G. Nasri-Roudsari, J. Rehberg, A note on the analytic form of the Kramer sampling theorem, Results Math. 34 (1998), no. 3-4, 310-319. MR 1652716 (99h:30036)
  • 3. G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge University Press, Cambridge, UK, 1990. MR 1052153 (91d:33034)
  • 4. G. H. Hardy, Notes on special systems of orthogonal functions (II): On functions orthogonal with respect to their own zeros, J. Lond. Math. Soc. 14 (1939), 37-44.
  • 5. J. R. Higgins, An interpolation series associated with the Bessel-Hankel transform, J. Lond. Math. Soc. 5 (1972), 707-714. MR 0320616 (47:9152)
  • 6. M. E. H. Ismail, The Zeros of Basic Bessel functions, the functions $ J_{\nu +ax}(x)$, and associated orthogonal polynomials, J. Math. Anal. Appl. 86 (1982), 1-19. MR 0649849 (83c:33010)
  • 7. M. E. H. Ismail, Some properties of Jackson's third q-Bessel function, unpublished manuscript.
  • 8. H. T. Koelink, The quantum group of plane motions and the Hahn-Exton q-Bessel function, Duke Math. J. 76 (1994), no. 2, 483-508. MR 1302322 (96a:33023)
  • 9. H. T. Koelink, R. F. Swarttouw, On the zeros of the Hahn-Exton q-Bessel Function and associated q-Lommel polynomials, J. Math. Anal. Appl. 186 (1994), 690-710. MR 1293849 (95j:33050)
  • 10. T. H. Koornwinder, R. F. Swarttouw, On q-analogues of the Fourier and Hankel transforms, Trans. Amer. Math. Soc. 333 (1992), no. 1, 445-461. MR 1069750 (92k:33013)
  • 11. H. P. Kramer, A generalized sampling theorem, J. Math. Phys. 38 1959/60, 68-72. MR 0103786 (21:2550)
  • 12. R. F. Swarttouw, H. G Meijer, A q-analogue of the Wronskian and a second solution of the Hahn-Exton q-Bessel difference equation. Proc. Amer. Math. Soc. 120 (1994), no. 3, 855-864. MR 1180467 (94e:33034)
  • 13. J. M. Whittaker, Interpolatory function theory (1935).

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Additional Information

Luis Daniel Abreu
Affiliation: Department of Mathematics, Universidade de Coimbra, Coimbra, Portugal 3001-454

Keywords: $q$-difference equations, $q$-Bessel functions, $q$-integral.
Received by editor(s): October 19, 2004
Received by editor(s) in revised form: April 7, 2005
Published electronically: March 23, 2006
Additional Notes: Partial financial assistance from Centro de Matemática da Universidade de Coimbra
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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