On Basic Fourier-Bessel Expansions

Abstract

When dealing with Fourier expansions using the third Jackson (also known as Hahn-Exton) q-Bessel function, the corresponding positive zeros jkν and the "shifted" zeros, qjkν, 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.