Fractional proportional differences with memory

Abstract

In this paper, we formulate nabla fractional sums and differences and the discrete Laplace transform on the time scale hℤ. Based on a local type h-proportional difference (without memory), we generate new types of fractional sums and differences with memory in two parameters which are generalizations to the formulated fractional sums and differences. The kernel of the newly defined generalized fractional sum and difference operators contain h-discrete exponential functions. The discrete h-Laplace transform and its convolution theorem are then used to study the newly introduced discrete fractional operators and also used to solve Cauchy linear fractional difference type problems with step 0 < h ≤ 1.