On Basic Fourier–Bessel Expansions

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.