On Basic Fourier-Bessel Expansions

[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.

[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