The author introduces in the -order fractional sum of as
and the -order fractional difference as
where is given and is such that . In Section 2, the basic properties of the fractional sum and difference operators are established. Section 3 is devoted to the study of composition rules for these operators (all possible combinations are discussed). In Section 4, the existence of the solution for the -order fractional difference equation
and the uniqueness of the solution for the latter equation completed with the initial values
are established. Finally, the developed theory is illustrated by two examples.