10.46298/hrj.2008.162 Bundschuh, Peter Peter Bundschuh Arithmetical investigations of particular Wynn power series episciences.org 2008 similar questions in other number fields irrationality Borwein's analytic method Wynn power series meromorphic continuation [MATH] Mathematics [math] contact@episciences.org episciences.org 2015-03-03T16:14:00+01:00 2021-05-28T00:30:19+02:00 2008-01-01T08:00:00+01:00 en Journal article https://hrj.episciences.org/162 https://hal.archives-ouvertes.fr/hal-01112326v1 PDF 1 Hardy-Ramanujan Journal ; Volume 31 - 2008 International audience Using Borwein's simple analytic method for the irrationality of the $q$-logarithm at rational points, we prove a quite general result on arithmetic properties of certain series, where the entering parameters are algebraic numbers. More precisely, our main result says that $\sum_{k\ge1}\beta^k/(1-\alpha q^k)$ is not in $\mathbb{Q}(q)$, if $q$ is an algebraic integer with all its conjugates (if any) in the open unit disc, if $\alpha\in\mathbb{Q}(q)^\times\setminus\{q^{-1},q^{-2},\ldots\}$ satisfies a mild denominator condition (implying $|q|>1$), and if $\beta$ is a unit in $\mathbb{Q}(q)$ with $|\beta|\le1$ but no other conjugates in the open unit disc. Our applications concern meromorphic functions defined in $|z|<|u|^{a\ell}$ by power series $\sum_{n\ge1}z^n/(\prod_{0\le\lambda<\ell}R_{a(n+\lambda)+b})$, where $R_m:=gu^m+hv^m$ with non-zero $u,v,g,h$ satisfying $|u|>|v|, R_m\ne0$ for any $m\ge1$, and $a,b+1,\ell$ are positive rational integers. Clearly, the case where $R_m$ are the Fibonacci or Lucas numbers is of particular interest. It should be noted that power series of the above type were first studied by Wynn from the analytical point of view.