Hahn introduced the difference operator D q,ω f (t) = (f (qt + ω) -f (t))/(t(q -1) + ω) in 1949, where 0 < q < 1 and ω > 0 are fixed real numbers. This operator extends the classical difference operator ω f (t) = (f (t + ω) -f (t))/ω as the Jackson q-differenceIn this paper, we present new results of the calculus based on the Hahn difference operator. Also, we establish an existence and uniqueness result of solutions of Hahn difference equations by using the method of successive approximations.