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SOME EXACT ALGEBRAIC EXPRESSIONS FOR THE TAILS OF TASOEV CONTINUED FRACTIONS

  • TAKAO KOMATSU (a1)
Abstract
Abstract

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:=qnαpn 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,…].

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References
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[1]Burger E. B., Exploring the Number Jungle: A Journey into Diophantine Analysis, Student Mathematical Library, 8 (American Mathematical Society, Providence RI, 2000).
[2]Cohn H., ‘A short proof of the simple continued fraction expansion of e’, Amer. Math. Monthly 113 (2006), 5762.
[3]Komatsu T., ‘On Tasoev’s continued fractions’, Math. Proc. Cambridge Philos. Soc. 134 (2003), 112.
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[5]Komatsu T., ‘Tasoev’s continued fractions and Rogers–Ramanujan continued fractions’, J. Number Theory 109 (2004), 2740.
[6]Komatsu T., ‘Hurwitz and Tasoev continued fractions’, Monatsh. Math. 145 (2005), 4760.
[7]Komatsu T., ‘An algorithm of infinite sums representations and Tasoev continued fractions’, Math. Comp. 74 (2005), 20812094.
[8]Komatsu T., ‘Hurwitz continued fractions with confluent hypergeometric functions’, Czechoslovak Math. J. 57 (2007), 919932.
[9]Komatsu T., ‘Tasoev continued fractions with long period’, Far East J. Math. Sci. (FJMS) 28 (2008), 89121.
[10]Komatsu T., ‘More on Hurwitz and Tasoev continued fractions’, Sarajevo J. Math. 4 (2008), 155180.
[11]Komatsu T., ‘A diophantine approximation of e 1/s in terms of integrals’, Tokyo J. Math. 32 (2009), 159176.
[12]Komatsu T., ‘Diophantine approximations of tanh, tan, and linear forms of e in terms of integrals’, Rev. Roumaine Math. Pures Appl. 54 (2009), 223242.
[13]Matala-Aho T. and Merilä V., ‘On Diophantine approximations of Ramanujan type q-continued fractions’, J. Number Theory 129 (2009), 10441055.
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[16]Perron O., Die Lehre von den Kettenbrüchen, Band I (Teubner, Stuttgart, 1954).
[17]Tasoev B. G., ‘Rational approximations to certain numbers’, Mat. Zametki 67 (2000), 931937 (Russian), English transl. in Math. Notes 67 (2000), 786–791.
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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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