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

Volume 167 / 2015

Peter Bundschuh, Keijo Väänänen Acta Arithmetica 167 (2015), 239-249 MSC: Primary 12F20; Secondary 11J91. DOI: 10.4064/aa167-3-2


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 $\mathbb {Q}$.


  • Peter BundschuhMathematisches Institut
    Universität zu Köln
    Weyertal 86-90
    50931 Köln, Germany
  • Keijo VäänänenDepartment of Mathematical Sciences
    University of Oulu
    P.O. Box 3000
    90014 Oulu, Finland

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