On algebraic independence of some continued fractions
Resumo
In the present paper, we prove the algebraic independence of a finite number of real continued fractions which have partial quotients that increase rapidly. We then use a general Liouville criteria to justiy the algebraic independence of limits of some real series. We note that these results extend some work of Bundschuh and we use a new and simple method.
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Referências
Adams, W. W., The algebraic independence of certain Liouville continued fractions, Proc. Amer. Math. Soc., 95(4), 521-516, (1985).
Belhroukia K. and Kacha, A., Transcendence and approximation measure of continued fraction, Int. J. Open Problems Compt. Math., 11(4), 1-14, (2018).
Bundschuh,P., Transcendental continued fractions, J. Number Theory 18, 91-98, (1984).
Durand, A., Indépendance algébrique de nombres complexes et critères de transcendance, compositio. Math. 35, 259-267, (1977).
Kacha, A., Approximation algébrique de fractions continues, C R. Acad. Sci. Paris, t. 317, Série I, 17-20, (1993).
Karadeniz Gozeri, G., Ayetin Pekin and Adem Kilicman, On the transcendence of some power series, Adv. Differ. Equ 17, 1-8, (2013).
Lianxiang, W., p-adic continued fraction (II), Scientia Sinica Ser. A, 28(10), 1018-1028, (1985).
Liouville, J., Sur des classes très étendues de quantités dont la valeur n’est ni algébrique, C. R. Mat. Acad. Sci. Paris 18, 883-885, 910-911, (1844).
Lorentzen, L., Wadeland, H., Continued fractions with Applications, Elsevier Science Publishers, (1992).
Nettler, G., Transcendental continued fractions, J. Number Theory 13, 456-462, (1981).
Okano, T., A note on the transcendental continued fractions, Tokyo J. Math.,10(1), 151-156, (1987).
K. F. Roth, K. F., Rational approximations to algebraic numbers, Mathematika, Vol 2, 1-20, (1955).
T. Schneider, T., Uber p-adich Kettenbruche, Symposia math. Vol. IV, 181-189, (1976).
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