# Arithmetical investigations of particular Wynn power series

1 MI - Mathematisches Institut - Universität zu Köln

Abstract : 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|

Author: Peter Bundschuh -

Source: https://hal.archives-ouvertes.fr/