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