In this paper, we study the generalized translation operator associated with the deformed Hankel transform on R. Firstly, we prove the trigonometric form of the generalized translation operator. Next, we derive the positivity of this operator on a suitable space of even functions. Making use of the positivity of the generalized translation operator we introduce and we study the deformed Stockwell transform. Knowing the fact that the study of reproducing kernels theory are both theoretically interesting and practically useful, we investigate for this transform the general theory of reproducing kernels theory. In particular, we investigate some applications of the Tikhonov regularization for the generalized Sobolev spaces and we study some time-frequency concentration problems.