Quantitative irrationality for sums of reciprocals of Fibonacci and Lucas numbers

Matala-aho, Tapani; Prévost, Marc

doi:10.1007/s11139-006-6511-4

Cited by 11 publications

(7 citation statements)

References 10 publications

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“…In fact, his approach to analyze the Padé approximants to the series ∞ n=1 z n /R n , where the sequence (R n ) n∈N satisfies a second order recurrence relation, led to similar results in the case, where the rational field is replaced by any imaginary quadratic number field. More recently, Matala-aho and Prévost [12], [13] extended these quantitative investigations using Padé approximations for particular Heine q-series as constructed in closed form by Matala-aho [11].…”

Section: Introduction and Resultsmentioning

confidence: 99%

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Arithmetical investigations of particular Wynn power series

Bundschuh

2008

Hardy-Ramanujan Journal

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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.

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Section: Introduction and Resultsmentioning

confidence: 99%

“…Here we applied also (12) and (13). Notice that, in the last triple sum, the summation is over all (κ, λ, µ) ∈ N 3 0 with κ + λ + µ = N − 1.…”

Section: Proof Of Theoremmentioning

confidence: 99%

Arithmetical investigations of particular Wynn power series

Bundschuh

2008

Hardy-Ramanujan Journal

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“…For further results on cyclotomic polynomials and least common multiples, one is referred to [8] and [9].…”

Section: On Arithmetical Properties Of Certain Q-series 143mentioning

confidence: 99%

On Arithmetical Properties of Certain q-Series

Merilä¹

2009

Result. Math.

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“…In [14] Matala-aho and Prévost also considered quantities of the form (1.1). However, not all the numbers we prove to be irrational are covered by their result.…”

Section: Introductionmentioning

confidence: 99%