An uncertainty principle for the basic Bessel transform

The aim of this paper is to prove an uncertainty principle for the basic Bessel transform of order \(\alpha \geq -\frac{1}{2}\) . In order to obtain a sharp uncertainty principle, we introduce and study a generalized q-Bessel-Dunkl transform which is based on the q-eigenfunctions of the q-Dunkl operator newly given by: $$T_{\alpha,q}(f)(x)=D_{q}f(x)+\frac{[2\alpha +1]_{q}}{2q^{2\alpha +1}}\frac{f(x)-f(-x)}{x}.$$ In this work, we will follow the same steps of Fitouhi et al. (Math. Sci. Res. J., 2007) using the operator T α,q instead of the q-derivative.