Algebraic independence of the power series defined by blocks of digits
Let \(q\geq 2\) be an integer and let \(w\) be a block of digits in \(\{0,1,\dots, q-1\}\). For an integer \(n\), denote by \(e(w,n)\) the number of occurrences of \(w\) in the base \(q\) expansion of \(n\). Let \(f(w,z)= \sum_{n\geq 0} e(w,n)z^n\). The author gives necessary and sufficient conditions on \(w_1,\dots, w_m\) for the algebraic independence of the functions \(f(w_1,z)\dots f(w_m,z)\), or of their values of \(f(w_1,\alpha)\dots f(w_m,\alpha)\) at an algebraic point \(\alpha\) with \(0<|\alpha|<1\).
scientific article; zbMATH DE number 1347933