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Rational and irrational series consisting of special denominators. (English) Zbl 1023.11036

The author deals with series of the type \(\alpha= \sum_{k=1}^\infty \frac{b_k}{A^{2^k}+ B^{2^k}}\), where \(A\), \(B\) are algebraic numbers, \(1\leq|B|<|A|\), \(A^2\), \(B^2\) are positive integers and \((b_n)\) is a sequence of integers which is connected with the numbers \(A\), \(B\) by certain relations. The author gives some conditions for the rationality of \(\alpha\).

MSC:

11J72 Irrationality; linear independence over a field
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References:

[1] Bundschuh P., Petho A.: Zur transzendenz gewisser Reihen. Monatsh. Math. 104, (1987), 199-223. · Zbl 0601.10025
[2] Duverney D.: Sur les series de nombres rationnels a convergence rapide. C. R. Acad. Sci., ser. I, Math. 328, no.7, (1999), 553-556. · Zbl 0940.11027
[3] Erdos P., Straus E.G.: On the irrationality of certain Ahmes series. J. Indian. Math. Soc. 27, (1968), 129-133. · Zbl 0131.04902
[4] Nishioka K.: Mahler Functions and Transcendence. Lecture notes in mathematics 1631 Springer) · Zbl 0876.11034
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