On the Graf’s addition theorem for Hahn Exton q-Bessel function

It is well known that the generalized translation operator T associated with the Bessel function of the first kind is positive in the sense if f > 0, then Tf > 0. This property is easily seen when we write Tf as an integral representation with a kernel involving the area of some triangle ([1],[9]) and has several applications in many mathematical fields such that hypergroup structure, heat equation.... In 2002, the appropriated q-generalized translation for the q-Bessel Hahn Exton function was founded [3] and the problem of its positivity asked. In literature we meet many attempts to show this property in particular case, nerveless they are no definitive response at to day. In [2] the authors prove that for the q-cosinus , the correspondent q-translation operator is positive if q ∈ [0, q0] for some q0. In this work and owing a new formulation of the Graf’s addition theorem [7] we give an affirmative answer about this theme by a technic involving some inclusion of sets. To make this work self containing, we begin by the following preliminaries. Throughout this paper we consider 0 < q < 1 and we adopt the standard conventional notations of [4]. For complex a We put