In this paper we extend the notion of an α-family of maps to discrete systems defined by simple difference equations with the fractional Caputo difference operator.The equations considered are equivalent to maps with falling factorial-law memory which is asymptotically power-law memory.We introduce the fractional difference Universal,Standard,and Logistic α-Families of Maps and propose to use them to study general properties of discrete nonlinear systems with asymptotically power-law memory.