@article{Bouzeffour2026, 
author = {Fethi Bouzeffour},
title = {Positive convolution structures for    q-Bessel functions and a discrete deformation of the Bessel–Kingman hypergroup},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {6},
pages = {19058-19087},
keywords = {q-Bessel functions, hypergroups, q-convolution, Fourier analysis, spectral theory, basic hypergeometric series},
url = {https://www.sciopen.com/article/10.3934/math.2026777},
doi = {10.3934/math.2026777},
abstract = {We construct a positive convolution on the discrete    q-lattice        K    q    =      q                  Z              ∪  {  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.}
}