A Gauss–Kuzmin-type problem for a family of continued fraction expansions

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⩾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.