Transcendence results on the generating functions of the characteristic functions of certain self-generating sets
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(α),G(α⁴) are algebraically independent for any algebraic α with 0 < |α| < 1. 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) ∈ ℤ × ℕ with 0 < |a| < √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(z⁴) over ℂ(z) leading to the F-analogue of Adamczewski's above-mentioned theorem.