On Basic Fourier-Bessel Expansions

TL;DR

This paper investigates the asymptotic behavior of zeros and the Jackson $q$-Bessel function in basic Fourier-Bessel expansions, establishing new relations and a $q$-analogue of the Riemann-Lebesgue theorem.

Contribution

It introduces asymptotic relations between zeros and shifted zeros of the Jackson $q$-Bessel function and develops a $q$-analogue of the Riemann-Lebesgue theorem.

Findings

Asymptotic relations between zeros and shifted zeros of Jackson $q$-Bessel function

Asymptotic behavior of Jackson $q$-Bessel function on shifted zeros

A $q$-analogue of the Riemann-Lebesgue theorem

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.