## Transcendence results on the generating functions of the characteristic functions of certain self-generating sets

### Volume 162 / 2014

#### Abstract

This article continues two papers which recently appeared in this same journal. First, Dilcher and Stolarsky [140 (2009)] introduced two new power series, $F(z)$ and $G(z)$, related to the so-called Stern polynomials and having coefficients 0 and 1 only. Shortly later, Adamczewski [142 (2010)] proved, inter alia, that $G(\alpha ),G(\alpha ^4)$ are algebraically independent for any algebraic $\alpha $ with $0<|\alpha |<1$. \par Our first key result is that $F$ and $G$ have large blocks of consecutive zero coefficients. Then, a Roth-type argument shows that $F(a/b)$ and $G(a/b)$, for any $(a,b)\in \mathbb {Z}\times \mathbb {N}$ with $0<|a|<\sqrt {b}$, are transcendental but not U-numbers. Moreover, reasonably good upper bounds for the irrationality exponent of these numbers are obtained. Another main result for which an elementary (or *poor men's*) proof is presented concerns the algebraic independence of $F(z),F(z^4)$ over $\mathbb {C}(z)$ leading to the $F$-analogue of Adamczewski's above-mentioned theorem.