Transcendence results on the generating functions of the characteristic functions of certain self-generating sets, II
Volume 167 / 2015
Abstract
This article continues a previous paper by the authors. Here and there, the two power series $F(z)$ and $G(z)$, first introduced by Dilcher and Stolarsky and related to the so-called Stern polynomials, are studied analytically and arithmetically. More precisely, it is shown that the function field $\mathbb {C}(z)(F(z),F(z^4), G(z),G(z^4))$ has transcendence degree 3 over $\mathbb {C}(z)$. This main result contains the algebraic independence over $\mathbb {C}(z)$ of $G(z)$ and $G(z^4)$, as well as that of $F(z)$ and $F(z^4)$. The first statement is due to Adamczewski, whereas the second is our previous main result. Moreover, an arithmetical consequence of the transcendence degree claim is that, for any algebraic $\alpha $ with $0<|\alpha |<1$, the field $\mathbb {Q}(F(\alpha ),F(\alpha ^4),G(\alpha ),G(\alpha ^4))$ has transcendence degree 3 over $\mathbb {Q}$.
