In this paper we study the q-analogue of the j a Bessel function (see (1)) which results after minor changes from the so-called Exton function studied by Koornwinder and Swarttow. Our objective is first to establish, using only the q-Jackson integral and the q-derivative, some properties of this function with proofs similar to the classical case; second to construct the associated q-Fourier analysis which will be used in a coming work to construct the q-analogue of the Besselhypergroup.

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In this paper, we study in quantum calculus the correspondence between poles of the q-Mellin transform (see [A. Fitouhi, N. Bettaibi, K. Brahim, The Mellin transform in Quantum Calculus, Constr. Approx. 23 (3) (2006) 305-323]) and the asymptotic behaviour of the original function at 0 and ∞. As applications, we give a new technique (in q-analysis) to derive the asymptotic expansion of some functions defined by q-integrals or by q-harmonic sums. Finally, a q-analogue of the Mellin-Perron formula is given.

This paper aims to study the q-wavelets and the q-wavelet transforms, using only the q-Jackson integrals and the q-cosine Fourier transform, for a fix q ∈]0, 1[. For this purpose, we shall attempt to extend the classical theory by giving their q-analogues.

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