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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.

Different type kernel<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mrow><mml:mi>h</mml:mi><mml:mo>−</mml:mo></mml:mrow></mml:math>fractional differences and their fractional<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mrow><mml:mi>h</mml:mi><mml:mo>−</mml:mo></mml:mrow></mml:math>sums

Different type kernelh−fractional differences and their fractionalh−sums