Transcendence of certain series involving binary linear recurrences
Let \(r>1\) be an integer, \((a_{k})_{k\geq 0}\) a sequence of rational integers such that \(\log |a_{k}|=o(r^k)\) and \(a_{k}\not=0\) for infinitely many \(k\). An example of the theorems which are proved in the paper under review is that the number \[ \sum_{k\geq 0} a_{k}/(2^{r^{k}}+3^{r^{k}}) \] is transcendental. The main result gives the arithmetic nature of the sum of series \(\sum_{k\geq 0} a_{k}/R_{cr^{k}+d}\), where, more generally, \((a_{k})_{k\geq 0}\) is a sequence of algebraic integers in a number field and \((R_{n})_{n\geq 0}\) a binary linear recurrent sequence \(R_{n+2}=A_{1}R_{n+1}+A_{2}R_{n}\) with \(A_{1}\) and \(A_{2}\) are in \(\mathbb Z\). The proof is an extension in several variables of a method by [\textit{D.~Duverney} and \textit{K.~Nishioka}, Acta Arith. 110, No. 4, 305--330 (2003; Zbl 1049.11078)], who studied the transcendence of the sum of series \(\sum_{k\geq 0} E_{k}(\alpha^{r^{k}})/F_{k}(\alpha^{r^{k}})\), where \((E_{k})_{k\geq 0}\) and \((F_{k})_{k\geq 0}\) are sequences of polynomials and \(\alpha\) is an algebraic number.
scientific article; zbMATH DE number 5129744