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]
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2021-05-28T00:30:19+02:00
2008-01-01T08:00:00+01:00
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Journal article
https://hrj.episciences.org/162
https://hal.archives-ouvertes.fr/hal-01112326v1
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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.