α-expansions, linear recurrences, and the sum-of-digits function

We give a generalization of the ordinary q-ary digit expansion of positive integers to arbitrary real bases α>1. We investigate the properties of the sum-of-digit function s(n). We give an asymptotic formula for the sum ∑ _n<N s(itn) and consider the distribution properties of the sequence xs(n) for irrational x.