We construct a positive convolution on the discrete $q$-lattice $K_q=q^{\mathbb{Z}}\cup \{0\}$ whose characters are the normalized Hahn–Exton $q$-Bessel functions. The convolution is obtained from a product formula arising as a limit of the Koelink–Floris product formula for little $q$-Jacobi polynomials, and its kernel is proved to be nonnegative and probability-preserving. The resulting structure gives a discrete $q$-deformation of the Bessel–Kingman hypergroup, but its convolution supports are generally noncompact and therefore lie outside the classical DJS axioms. We introduce a degenerate DJS framework adapted to this setting and prove the corresponding Fourier inversion, Plancherel formula, and spectral decomposition for the $q$-Bessel operator.

In this work, we develop the harmonic analysis associated with the second-order differential operator ${\mathcal{L}}_{\gamma }=-\frac{d^2}{d{x}^2}-2\gamma \frac{d}{dx}-{\gamma }^2$. Fractional powers of ${\mathcal{L}}_{\gamma }$ are defined via spectral representation, and a singular integral representation is provided. Furthermore, we establish the equivalence between the fractional powers of ${\mathcal{L}}_{\gamma }$ and a tempered Riesz derivative.