Ramanujan J (2009) 18: 171–182 DOI 10.1007/s11139-007-9117-6 An uncertainty principle for the basic Bessel transform Ahmed Fitouhi · Néji Bettaibi · Wafa Binous · Hédi Ben Elmonser Received: 3 January 2007 / Accepted: 18 December 2007 / Published online: 17 December 2008 © Springer Science+Business Media, LLC 2008 Abstract The aim of this paper is to prove an uncertainty principle for the basic Bessel transform of order α ≥− 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 α,q (f )(x) = D q f(x) + [2α + 1] q 2q 2α+1 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. Keywords Quantum calculus · q -Jackson integrals · q -Bessel function · q -Bessel transform · q -Dunkl operator Mathematics Subject Classification (2000) Primary 33D15 · Secondary 33D60 · 42A38 · 47A63 · 26D15 · 47B38 A. Fitouhi ( ) Faculté des Sciences de Tunis, 1060 Tunis, Tunisia e-mail: Ahmed.Fitouhi@fst.rnu.tn N. Bettaibi Institut Préparatoire aux Études d’Ingénieur de Nabeul, 8000 Nabeul, Tunisia e-mail: Neji.Bettaibi@ipein.rnu.tn W. Binous Institut Biotechnologie de Béja, 9000 Béja, Tunisia e-mail: wafa.binous@gnet.tn H.B. Elmonser Institut Préparatoire aux Études d’Ingénieur de Bizerte, Bizerte, Tunisia