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

Fethi Bouzeffour; Fethi Bouzeffour

doi:10.3934/math.2026777

AIMS Mathematics

2026, Volume 11, Issue 6: 19058-19087. doi: 10.3934/math.2026777

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

Fethi Bouzeffour ^,

Department of Mathematics, College of Sciences, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Received: 25 April 2026 Revised: 24 June 2026 Accepted: 25 June 2026 Published: 29 June 2026
MSC : 33D15, 33D45, 43A62, 43A32, 47A10

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.
- $ q $-Bessel functions,
- hypergroups,
- $ q $-convolution,
- Fourier analysis,
- spectral theory,
- basic hypergeometric series
Citation: Fethi Bouzeffour. Positive convolution structures for $ q $-Bessel functions and a discrete deformation of the Bessel–Kingman hypergroup[J]. AIMS Mathematics, 2026, 11(6): 19058-19087. doi: 10.3934/math.2026777

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Abstract

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.

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