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Quantitative irrationality for sums of reciprocals of Fibonacci and Lucas numbers. (English) Zbl 1104.11040

Let \(W_n=A \alpha ^n+B\beta ^n\) be either the Fibonacci or the Lucas sequence. Then under special conditions the authors prove that \[ \frac{\log (|\alpha|/|\beta|)}{\log (|\alpha|^{c-1}/|\beta|)} \] is an upper bound of the irrationality measure for the sum of the series \(\sum_{n=0}^\infty \frac{t^n}{W_{an+b}}\). Here \[ c=\frac 12 +\Bigl(\frac{\phi(a)}{2a^2}-\frac 3{\pi^2}\sum_{l=1, (l,a)=1}^{a-1}\frac 1{l^2}\Bigr)\prod_{p\mid a} \frac{p^2}{p^2-1} \] for the Fibonacci sequence and \[ c=\frac 12 +\Bigl(\frac{\phi(a)\gcd(a,2)^2}{6a^2}-\frac 4{\pi^2} \sum_{l=1, (l,2a)=1}^{a-1}\frac 1{l^2}\Bigr)\prod_{p\mid a, p\geq 3} \frac{p^2}{p^2-1} \] for the Lucas sequence where \(\varphi (x)\) is the Euler phi-function.

MSC:

11J82 Measures of irrationality and of transcendence
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
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