  In this paper we study in detail a family of continued fraction expansions of
any number in the unit closed interval $[0,1]$ whose digits are differences of
consecutive non-positive integer powers of an integer $m \geq 2$. For the
transformation which generates this expansion and its invariant measure, the
Perron-Frobenius operator is given and studied. For this expansion, we apply
the method of random systems with complete connections by Iosifescu and
obtained the solution of its Gauss-Kuzmin type problem.
