•The fractional differences and sums within delta and nabla are studied summarized and related by some identities so that it becomes helpful to proceed to other types of fractional differences.•The definition of left and right fractional differences with discrete exponential kernels and their corresponding sums have been given and studied on the time scale hZ. (CFC and CFR h-fractional differences and their corresponding CF h-fractional sums)•The definition of left and right fractional differences with discrete MittagLeffler kernels and their corresponding sums have been given and studied on the time scale hZ. (ABC and ABR h-fractional differences and their corresponding AB h-fractional sums)•Working on hZ will guarantee better numerical solutions for complex system models depending on discrete fractional calculus and it will affect the domain of convergence for the fractional differences and sums. The aim of this article is to recall and study fractional derivatives with singular kernels on hZ and define fractional derivatives with non-singular exponential and Mittag–Leffler kernels on hZ and study some of their properties. We shall follow the nabla time scale analysis and relate the h−nabla classical discrete fractional derivatives to the delta existing ones studied before by some authors. Some dual identities between left and right and delta and nabla, left and right h−fractional difference types will be investigated. The nabla h− discrete versions of the Mittag-Leffler functions will be recalled by means of the nabla h−fatorial functions and nabla h−Taylor polynomials. The discrete Laplace on hZ and its convolution theory are used often to proceed in our investigation. The obtained results will generalize the nabla classical discrete fractional differences and the nabla discrete fractional differences with discrete exponential and ML−kernels studied recently by Abdeljawad and Baleanu by setting h=1.