  In the recent development in a various disciplines of physics, it is noted
the need for including the deformed versions of the exponential functions. In
this paper, we consider the deformations which have two purposes: to have them
like special cases, and, even more, to acquit their inauguration from
mathematical point of view. Really, we will show interesting differential and
difference properties of our deformations which are important in forming and
explanation of continuous and discrete models of numerous phenomena.
