Exponential Functions and Laplace Transforms for Alpha Derivatives

We introduce the exponential function for alpha derivatives on generalized time scales. We also define the Laplace transform that helps to solve higher order linear alpha dynamic equations on generalized time scales. If # = #, the Hilger forward jump operator, then our theory contains the theory of delta dynamic equations on time scales as a special case. If # = #, the Hilger backward jump operator, then our theory contains the theory of nabla dynamic equations on time scales as a special case. Hence di#erential equations, di#erence equations (using the forward or backward di#erence operator) , or q-di#erence equations (using the forward or backward q-di#erence operator) can be accommodated within our theory. We also present various properties of the Laplace transform and o#er some examples