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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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 | References | Similar Articles | Additional Information

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.


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Additional Information

Luis Daniel Abreu
Affiliation: Department of Mathematics, Universidade de Coimbra, Coimbra, Portugal 3001-454
Email: daniel@mat.uc.pt

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08285-2
PII: S 0002-9939(06)08285-2
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.