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Remarks on irrationality of \(q\)-harmonic series. (English) Zbl 1044.11068
Given some \(p\in\mathbb{Z}\setminus\{0, \pm 1\}\), we may consider the two series \[ h_p(1)=\sum_{\nu=1}^{\infty} \frac{1}{p^{\nu}-1}\quad \text{and} \quad \ln_p(2)=\sum_{\nu=1}^{\infty} \frac{(-1)^{\nu-1}}{p^{\nu}-1}. \] The author improves the best currently known upper bound for the irrationality measure \(\mu\) of \(h_p(1)\) and \(\ln_p(2)\). He proves that for all \(p\in\mathbb{Z}\setminus\{0, \pm 1\}\): \[ \mu(h_p(1))\leq 2.49846482... \quad \text{and} \quad \mu(\ln_p(2))\leq 3.29727451... \] To prove these results, he first introduces a basic hypergeometric series with a particular form which enables him to construct rational linear forms in 1 and \(h_p(1)\), resp. 1 and \(\ln_p(2)\). He then adapts the (by now well-known) methods used to deal with ordinary hypergeometric series. In particular, he adapts a classical process used to extract “big” common factors from the coefficients of the rational linear forms .

MSC:
11J82 Measures of irrationality and of transcendence
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
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