Transcendence of certain series involving binary linear recurrences
type: Item
instance of: scholarly article
title: Transcendence of certain series involving binary linear recurrences
zbMATH Open document ID: 1109.11035
author: Takeshi Kurosawa
published in: Journal of Number Theory
publication date: 2007-02-26T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 11J81
zbMATH DE Number: 5129744
zbMATH Keywords: transcendence
zbMATH Keywords: binary linear recurrences
reviewed by: Michel Waldschmidt
MaRDI profile type: Publication
full work available at URL: j.jnt.2006.05.019
OpenAlex ID: W2063815666
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DOI: 10.1016/J.JNT.2006.05.019
description: scientific article; zbMATH DE number 5129744