A Gauss-Kuzmin Theorem for Some Continued Fraction Expansions

Abstract

We consider 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 this expansion, we apply the method of Rockett and Sz\"usz from [6] and obtained the solution of its Gauss-Kuzmin type problem.