Tokyo Journal of Mathematics

Irrationality of Certain Lambert Series

Yohei TACHIYA

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Abstract

Let $q$ with $|q|>1$ be an integer in an algebraic number field $\mathbf{K}$ given below, and $\{a_n\}$ a periodic sequence in $\mathbf{K}$ of period two, not identically zero. Let $f(q)=\sum_{n=1}^{\infty}{a_n}/(1-q^n).$ We prove that (i) If $\mathbf{K}$ is either the rational number field $\mathbb{Q}$ or an imaginary quadratic field, then $f(q)\notin \mathbf{K}$. (ii) For an algebraic integer $q$ such that $|q|>1$ and $|q^{\sigma}|<1$ for any $\sigma \in$ $\textrm{Aut}(\overline{\mathbb{Q}}/\mathbb{Q})$ with $q^\sigma\ne q$, if $k=\mathbb{Q}(q)$, then $f(q)\notin \mathbb{Q}(q)$. For example, the three numbers $$ 1\,,\quad \sum_{n=1}^{\infty}\frac{1}{q^n-1}\,, \quad \sum_{n=1}^{\infty}\frac{1}{q^n+1} $$ are linearly independent over $\mathbb{Q}$ for every $q\in\mathbb{Z}$ with $|q|\geq2.$ Further, irrationality results of the special values of the functions $$ \sum_{n=1}^{\infty} \frac{z^n}{R_{an+b}}\,,\qquad \sum_{n=1}^{\infty}\frac{z^n}{R_{an+b}R_{{a(n+1)}+b}}~~(z\in\mathbb{C}) $$ can be deduced, where $a>0$, $b\geq0$ are integers and $R_n$ is a certain binary recurrence.

Article information

Source
Tokyo J. Math., Volume 27, Number 1 (2004), 75-85.

Dates
First available in Project Euclid: 5 June 2009

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1244208475

Digital Object Identifier
doi:10.3836/tjm/1244208475

Mathematical Reviews number (MathSciNet)
MR2060075

Zentralblatt MATH identifier
1072.11051

Subjects
Primary: 11J72: Irrationality; linear independence over a field

Citation

TACHIYA, Yohei. Irrationality of Certain Lambert Series. Tokyo J. Math. 27 (2004), no. 1, 75--85. doi:10.3836/tjm/1244208475. https://projecteuclid.org/euclid.tjm/1244208475


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