Positive convolution structures for $q$-Bessel functions and a discrete deformation of the Bessel–Kingman hypergroup

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.