  We Investigate two types of dual identities for Caputo fractional
differences. The first type relates nabla and delta type fractional sums and
differences. The second type represented by the Q-operator relates left and
right fractional sums and differences. Two types of Caputo fractional
differences are introduced, one of them (dual one) is defined so that it obeys
the investigated dual identities. The relation between Rieamnn and Caputo
fractional differences is investigated and the delta and nabla discrete
Mittag-Leffler functions are confirmed by solving Caputo type linear fractional
difference equations. A nabla integration by parts formula is obtained for
Caputo fractional differences as well.
