We will present a summary of the latest developments with respect to expansions involving the Third Jackson q-Bessel function (or, Hahn-Exton q-Bessel function), with special emphasis regarding Fourier expansions related to some basic functions which are q-analogues of the classical trigonometric and Bessel functions. Topics such as, properties of the analogues of the Lebesgue spaces associated with the Jackson q-integral and the Jackson-Thomae (or, Jackson-Nörlund) (q,w)-integral, as well as Orthogonality, Completeness, Convergence and the Riemann-Lebesgue behavior of the Fourier coefficients, will be detailed.