Amdeberhan, Tewodros; Zeilberger, Doron \(q\)-Apéry irrationality proofs by \(q\)-WZ pairs. (English) Zbl 0914.11042 Adv. Appl. Math. 20, No. 2, 275-283 (1998). The authors improve the irrationality measure of \[ \sum^\infty_{k=1} {1\over q^k- 1}\quad\text{and} \quad \sum^\infty_{n=1} {(1)^n\over q^n- 1} \] (\(q\) is an integer, \(| q|>1\)) to 4.80. The proofs use the so-called Apéry method of the irrationality of \(\zeta(3)\). Reviewer: J.Hančl (Ostrava) Cited in 1 ReviewCited in 5 Documents MSC: 11J82 Measures of irrationality and of transcendence 11J72 Irrationality; linear independence over a field Keywords:irrationality measure; Apéry method PDFBibTeX XMLCite \textit{T. Amdeberhan} and \textit{D. Zeilberger}, Adv. Appl. Math. 20, No. 2, 275--283 (1998; Zbl 0914.11042) Full Text: DOI arXiv References: [1] Apéry, R., Irrationalitè de ζ(2) et ζ(3), Asterisque, 61, 11-13 (1979) · Zbl 0401.10049 [2] Borwein, P., On the irrationality of ∑1/(\(q^n}+r\), J. Number Theory, 37, 253-259 (1991) · Zbl 0718.11029 [3] Borwein, P., On the irrationality of certain series, Proc. Cambridge Philos. Soc., 112, 141-146 (1992) · Zbl 0779.11027 [4] Erdős, P., On arithmetical properties of Lambert Series, J. Indian Math. Soc. (N.S.), 12, 63-66 (1948) · Zbl 0032.01701 [5] Erdős, P., On the irrationality of certain series: problems and results, New Advances in Transcendence Theory (1988), Cambridge University Press: Cambridge University Press Cambridge, p. 102-109 [6] van der Poorten, A., A proof that Euler missed…, Apéry’s proof of the irrationality of ζ(3), Math. Intelligence, 1, 195-203 (1979) · Zbl 0409.10028 [7] Petkovšek, M.; Wilf, H. S.; Zeilberger, D., \(AB (1996)\), A. K. Peters [8] Zeilberger, D., Closed form (pun intended!), Contemp. Math., 143, 579-607 (1993) · Zbl 0808.05010 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.