On Basic Fourier-Bessel Expansions

Symmetry Integrability and Geometry-Methods and Applications

article

José Luis Cardoso¹ 

[1] Mathematics Department, University of Trás-os-Montes e Alto Douro (UTAD)

关键词: third Jackson q-Bessel function;    Hahn–Exton q-Bessel function;    basic Fourier– Bessel expansions;    basic hypergeometric function;    asymptotic behavior;    Riemann–Lebesgue theorem;   

DOI  :  10.3842/SIGMA.2018.035

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown