Arithmetical investigations of particular Wynn power series

Abstract

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.

Series of reciprocal products with factors from linear recurrence sequences

“…We now state our main result, from which we will deduce later, by suitable specialization, arithmetic data on Melham's series (3). But first we recall that the denominator of an algebraic number d, denoted by denðdÞ, is the smallest d a N such that d Á d is an algebraic integer.…”

“…The main aim of the present paper is to investigate arithmetically series of type (3), in fact, far-reaching generalizations of them, without recourse to their representation by the above-mentioned Lambert series. Plainly, for that purpose we need arithmetical hypotheses on the parameter p in (1).…”

“…on which it was shown in Lemma 3 of [3]: The series (10), convergent in jzj < jaj k , has a meromorphic continuation to the whole complex plane. This continuation Wðz; k; 'Þ has its poles exactly at the points a k À a b Á ik , i a N 0 , and they are all simple.…”

“…Clearly, ( 14) describes the connection of the function W j ðz; k; '; mÞ with Wðz; k; 'Þ. Having all analytic tools for the proof of our theorem, we next quote Theorem 2 from [3] as our main arithmetic tool. hold for any s a AutðQjQÞnfidg.…”

Linear independence results for certain sums of reciprocals of Fibonacci and Lucas numbers

“…Remark 1.1. In 1989, R. André-Jeannin [1] proved the irrationality of the fundamental sum f := ∞ n=1 1/F n ; see also [3,5,13]. More generally, P. Bundschuh and K. Väänänen [2] obtained f / ∈ Q( √ 5) as well as an irrationality measure.…”