The First and Second Translation Theorem for Laplace Transform on Time Scale

In our paper we formulate and prove the first and second translation theorem for time scale Laplace transform, for several elementary functions. We use the definition of Laplace transform on arbitrary time scales as introduced by M. Bohner and A. Peterson. For time scale taken to be the set of real numbers, the classical definition of Laplace transform is obtained. However, taking the set of integers for time scale, the modification of translation theorems for $\mathcal{Z}$-transform is obtained. This approach applies to all time scales and represents unification and extension of classical Laplace and $\mathcal{Z}$-transform, for specified functions.