Abstract
In this article, we solve the open problem 24.5.6 given in the study of Ismail, which consists of extending the action of
q
-translation operators introduced by Ismail to some measurable functions by means of basic Fourier theory. Also, we prove that the
q
-exponential function is the only solution of the
q
-analogue of the Cauchy functional equation. As application we give an inversion formula for the
q
-Gauss Weierstrass transform.
2000 Mathematics Subject Classification:
33D45; 33D60.