Discrete generalized fractional operators defined using h-discrete Mittag-Leffler kernels and applications to AB fractional difference systems

This study investigates the h-fractional difference operators with h-discrete generalized Mittag-Leffler kernels ( hE phi,delta?omega(Theta,t-rho h(sh)) in the sense of Riemann type (namely, the ABR) and Caputo type (namely, the ABC). For which, we will discuss the region of convergent. Then, we study the h-discrete Laplace transforms to formulate their corresponding AB-fractional sums. Also, it is useful in obtaining the semi-group properties. We will prove the action of fractional sums on the ABC type h-fractional differences and then it can be used to solve the system of ABC h-fractional difference. By using the h-discrete Laplace transforms and the Picard successive approximation technique, we will solve the nonhomogeneous linear ABC h-fractional difference equation with constant coefficient, and also we will remark the h-discrete Laplace transform method for the continuous counterpart. Meanwhile, we will obtain a nontrivial solution for the homogeneous linear ABC h-fractional difference initial value problem with constant coefficient for the case delta not equal 1. We will formulate the relation between the ABC and ABR h-fractional differences by using the h-discrete Laplace transform. By iterating the fractional sums of order -(phi, delta, 1), we will generate the h-fractional sum-differences, and in view of this, a semigroup property will be proved. Due to these new powerful techniques, we can calculate the nabla h-discrete transforms for the AB h-fractional sums and the AB iterated h-fractional sum-differences. Furthermore, we will obtain some particular cases that can be found in examples and remarks. Finally, we will discuss the higher order case of the h-discrete fractional differences and sums.