On the Laplace-Type Transform and Its Applications

Using the Laplace transform and the Gamma function, we obtain the Laplace-type transform, with the property of mapping a function to a functional sequence, which cannot be realized by the Laplace transform. In addition, we construct a backward difference as a generalization of the backward difference operator ∇. By connecting it to the Laplace-type transform, we deduce a method for solving difference equations and, relying on classical orthogonal polynomials, for obtaining combinatorial identities. A table of some elementary functions and their images is at the end of the text.