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SOME EXACT ALGEBRAIC EXPRESSIONS FOR THE TAILS OF TASOEV CONTINUED FRACTIONS

Part of: Diophantine approximation, transcendental number theory Elementary number theory

Published online by Cambridge University Press:  11 October 2012

TAKAO KOMATSU
TAKAO KOMATSU*
Affiliation:
Graduate School of Science and Technology, Hirosaki University, Hirosaki 036-8561, Japan (email: komatsu@cc.hirosaki-u.ac.jp)
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Abstract

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Denote the nth convergent of the continued fraction α=[a0;a1,a2,…] by pn/qn=[a0;a1,…,an]. In this paper we give exact formulae for the quantities Dn:=qnα−pn in several typical types of Tasoev continued fractions. A simple example of the type of Tasoev continued fraction considered is α=[0;ua,ua2,ua3,…].

Keywords

Tasoev continued fractionsHurwitz continued fractionscontinued fractionsDiophantine approximations

MSC classification

Secondary: 11A55: Continued fractions 11J70: Continued fractions and generalizations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 92 , Issue 2 , April 2012 , pp. 179 - 193
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

References

[1]Burger, E. B., Exploring the Number Jungle: A Journey into Diophantine Analysis, Student Mathematical Library, 8 (American Mathematical Society, Providence RI, 2000).Google Scholar
[2]Cohn, H., ‘A short proof of the simple continued fraction expansion of e’, Amer. Math. Monthly 113 (2006), 57–62.Google Scholar
[3]Komatsu, T., ‘On Tasoev’s continued fractions’, Math. Proc. Cambridge Philos. Soc. 134 (2003), 1–12.CrossRefGoogle Scholar
[4]Komatsu, T., ‘On Hurwitzian and Tasoev’s continued fractions’, Acta Arith. 107 (2003), 161–177.CrossRefGoogle Scholar
[5]Komatsu, T., ‘Tasoev’s continued fractions and Rogers–Ramanujan continued fractions’, J. Number Theory 109 (2004), 27–40.CrossRefGoogle Scholar
[6]Komatsu, T., ‘Hurwitz and Tasoev continued fractions’, Monatsh. Math. 145 (2005), 47–60.CrossRefGoogle Scholar
[7]Komatsu, T., ‘An algorithm of infinite sums representations and Tasoev continued fractions’, Math. Comp. 74 (2005), 2081–2094.CrossRefGoogle Scholar
[8]Komatsu, T., ‘Hurwitz continued fractions with confluent hypergeometric functions’, Czechoslovak Math. J. 57 (2007), 919–932.CrossRefGoogle Scholar
[9]Komatsu, T., ‘Tasoev continued fractions with long period’, Far East J. Math. Sci. (FJMS) 28 (2008), 89–121.Google Scholar
[10]Komatsu, T., ‘More on Hurwitz and Tasoev continued fractions’, Sarajevo J. Math. 4 (2008), 155–180.Google Scholar
[11]Komatsu, T., ‘A diophantine approximation of e 1/s in terms of integrals’, Tokyo J. Math. 32 (2009), 159–176.CrossRefGoogle Scholar
[12]Komatsu, T., ‘Diophantine approximations of tanh, tan, and linear forms of e in terms of integrals’, Rev. Roumaine Math. Pures Appl. 54 (2009), 223–242.Google Scholar
[13]Matala-Aho, T. and Merilä, V., ‘On Diophantine approximations of Ramanujan type q-continued fractions’, J. Number Theory 129 (2009), 1044–1055.CrossRefGoogle Scholar
[14]Mc Laughlin, J., ‘Some new families of Tasoevian and Hurwitzian continued fractions’, Acta Arith. 135 (2008), 247–268.CrossRefGoogle Scholar
[15]Osler, T., ‘A proof of the continued fraction expansion of e 1/M’, Amer. Math. Monthly 113 (2006), 62–66.Google Scholar
[16]Perron, O., Die Lehre von den Kettenbrüchen, Band I (Teubner, Stuttgart, 1954).Google Scholar
[17]Tasoev, B. G., ‘Rational approximations to certain numbers’, Mat. Zametki 67 (2000), 931–937 (Russian), English transl. in Math. Notes 67 (2000), 786–791.Google Scholar