Algebraic Independence of the Power Series Defined by Blocks of Digits

AbstractLet q⩾2 be an integer and let w be a block of 0, …, q−1 of finite length. For a nonnegative integer n, let e(w; n) denote the number of occurrences of w in the q-adic expansion of n. Define f(w; z)=∑n⩾0e(w; n)zn. We give necessary and sufficient conditions for the algebraic independence of functions f(w1; z), …, f(wm; z) and their values for given blocks w1, …, wm