### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 035, 13 pages      arXiv:1707.05216      https://doi.org/10.3842/SIGMA.2018.035

### On Basic Fourier-Bessel Expansions

José Luis Cardoso
Mathematics Department, University of Trás-os-Montes e Alto Douro (UTAD), Vila Real, Portugal

Received September 27, 2017, in final form April 11, 2018; Published online April 17, 2018

Abstract
When dealing with Fourier expansions using the third Jackson (also known as Hahn-Exton) $q$-Bessel function, the corresponding positive zeros $j_{k\nu}$ and the ''shifted'' zeros, $qj_{k\nu}$, among others, play an essential role. Mixing classical analysis with $q$-analysis we were able to prove asymptotic relations between those zeros and the ''shifted'' ones, as well as the asymptotic behavior of the third Jackson $q$-Bessel function when computed on the ''shifted'' zeros. A version of a $q$-analogue of the Riemann-Lebesgue theorem within the scope of basic Fourier-Bessel expansions is also exhibited.

Key words: third Jackson $q$-Bessel function; Hahn-Exton $q$-Bessel function; basic Fourier-Bessel expansions; basic hypergeometric function; asymptotic behavior; Riemann-Lebesgue theorem.

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