  Despite many applications regarding fractional calculus have been reported in
literature, it is still unknown how to model some practical process. One major
challenge in solving such a problem is that, the nonlocal property is needed
while the infinite memory is undesired. Under this context, a new kind nabla
fractional calculus accompanied by a tempered function is formulated. However,
many properties of such fractional calculus needed to be discovered. From this,
this paper gives particular emphasis to the topic. Some remarkable properties
like the equivalence relation, the nabla Taylor formula, and the nabla Laplace
transform for such nabla fractional calculus are developed and analyzed. It is
believed that this work greatly enriches the mathematical theory of nabla
tempered fractional calculus and provides high value and huge potential for
further applications.
