Irrationality measures for the series of reciprocals from recurrence sequences.
The paper under review deals with rational approximation (or more precisely, approximations by quotients of integers in an imaginary quadratic field) for complex numbers of the shape \(\sum_{n=0}^\infty t^n/W_n\), where \(W_n\) satisfies a second-order recurrence \(W_{n+2}= rW_{n+1}+sW_n\), (\(r,s,W_1,W_2\in\mathbb{Q}^*\) and \(t\) lying in a subset of an imaginary quadratic field. Using Padd approximations of Heine's \(q\)-hypergeometric series, the authors obtain upper bounds (and this, uniformly with respect to \(t\)) for the irrationality measure of the values of such series. In the particular cases where \((W_n)_{n>0}\) is the Fibonacci or the Lucas sequence, the bounds are \(\frac{2\pi^2}{\pi^2-3} =2,87341\dots\) and \(\frac{3\pi^2}{\pi^2-6}=7,65163\dots\), respectively. This substantially improves earlier result's obtained for these two sequences.
scientific article; zbMATH DE number 1886719
scientific article