Algebraic independence of the generating functions of Stern's sequence and of its twist
The sequence \((a_0,a_1,a_2,\dots)\) defined by \(a_0=0\), \(a_1=1\), \(a_{2n}=a_n\) and \(a_{2n+1}=a_n+a_{n+1}\) for \(n\geq 1\) was introduced by \textit{M. A. Stern} [J. Reine Angew. Math. 55, 193--220 (1858; ERAM 055.1457cj)]. The generating function \(A(z)=\sum_{n\geq 0} a_{n+1}z^n\) satisfies \(A(z)=p(z)A(z^2)\) and \(A(z)=\prod_{k=0}^\infty p(z^{2^k})\) with \(p(z)=1+z+z^2\). A twisted version \((b_0,b_1,b_2,\dots)\) was introduced by \textit{R. Bacher} [``Twisting the Stern sequence'', \url{arXiv:1005.5627}] and \textit{J.-P. Allouche} [``On the Stern sequence and its twisted version'', Integers 12, A58 (2012), \url{http://arxiv.org/abs/1202.4171}], where \(b_0=0\), \(b_1=1\), \(b_{2n}=-b_n\) and \(b_{2n+1}=-b_n-b_{n+1}\) for \(n\geq 1\). The generating function \(B(z)=\sum_{n\geq 0} b_{n+1}z^n\) satisfies \(B(z)=2-p(z)B(z^2)\). In [Int. J. Number Theory 6, No. 1, 211--217 (2010; Zbl 1250.11015)], \textit{M.~Coons} proved that \(A(\alpha)\) is transcendental for \(\alpha\) algebraic with \(0<|\alpha|<1\). In [Int. J. Number Theory 8, No. 2, 361--376 (2012; Zbl 1288.11070)], the first author proved that the numbers \(A(\alpha),A'(\alpha),A''(\alpha),\dots\) are algebraically independent for the same values of \(\alpha\). Here, the authors prove the same result for \(B\) in place of \(A\). Further, they prove that for any complex number \(\alpha\) with \(0<|\alpha|<1\), the transcendence degree over \(\mathbb Q\) of the field \(\mathbb Q(\alpha,A(\alpha),B(\alpha))\) is at least two. Furthermore, for algebraic \(\alpha\) with \(0<|\alpha|<1\), they produce a measure of algebraic independence of the two numbers \(A(\alpha)\), \(B(\alpha)\). Finally, for rational \(\alpha=r/s\) with \(0<|r|<s\), they produce irrationality measures for \(A(r/s)\) and for \(B(r/s)\).
scientific article; zbMATH DE number 6173996