Leaping convergents of Tasoev continued fractions

Takao Komatsu

Discussiones Mathematicae - General Algebra and Applications (2011)

  • Volume: 31, Issue: 2, page 201-216
  • ISSN: 1509-9415

Abstract

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Denote the n-th convergent of the continued fraction by pₙ/qₙ = [a₀;a₁,...,aₙ]. We give some explicit forms of leaping convergents of Tasoev continued fractions. For instance, [0;ua,ua²,ua³,...] is one of the typical types of Tasoev continued fractions. Leaping convergents are of the form p r n + i / q r n + i (n=0,1,2,...) for fixed integers r ≥ 2 and 0 ≤ i ≤ r-1.

How to cite

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Takao Komatsu. "Leaping convergents of Tasoev continued fractions." Discussiones Mathematicae - General Algebra and Applications 31.2 (2011): 201-216. <http://eudml.org/doc/276577>.

@article{TakaoKomatsu2011,
abstract = {Denote the n-th convergent of the continued fraction by pₙ/qₙ = [a₀;a₁,...,aₙ]. We give some explicit forms of leaping convergents of Tasoev continued fractions. For instance, [0;ua,ua²,ua³,...] is one of the typical types of Tasoev continued fractions. Leaping convergents are of the form $p_\{rn+i\}/q_\{rn+i\}$ (n=0,1,2,...) for fixed integers r ≥ 2 and 0 ≤ i ≤ r-1.},
author = {Takao Komatsu},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {leaping convergents; Tasoev continued fractions},
language = {eng},
number = {2},
pages = {201-216},
title = {Leaping convergents of Tasoev continued fractions},
url = {http://eudml.org/doc/276577},
volume = {31},
year = {2011},
}

TY - JOUR
AU - Takao Komatsu
TI - Leaping convergents of Tasoev continued fractions
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2011
VL - 31
IS - 2
SP - 201
EP - 216
AB - Denote the n-th convergent of the continued fraction by pₙ/qₙ = [a₀;a₁,...,aₙ]. We give some explicit forms of leaping convergents of Tasoev continued fractions. For instance, [0;ua,ua²,ua³,...] is one of the typical types of Tasoev continued fractions. Leaping convergents are of the form $p_{rn+i}/q_{rn+i}$ (n=0,1,2,...) for fixed integers r ≥ 2 and 0 ≤ i ≤ r-1.
LA - eng
KW - leaping convergents; Tasoev continued fractions
UR - http://eudml.org/doc/276577
ER -

References

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  1. [1] C. Elsner, On arithmetic properties of the convergents of Euler's number, Colloq. Math. 79 (1999), 133-145. Zbl0930.11048
  2. [2] C. Elsner, T. Komatsu and I. Shiokawa, Approximation of values of hypergeometric functions by restricted rationals, J. Théor. Nombres Bordeaux 19 (2007), 393-404. doi: 10.5802/jtnb.593 Zbl1167.11026
  3. [3] C. Elsner, T. Komatsu and I. Shiokawa, On convergents formed from Diophantine equations, Glasnik Mat. 44 (2009), 267-284. doi: 10.3336/gm.44.2.02 Zbl1233.11031
  4. [4] T. Komatsu, On Tasoev's continued fractions, Math. Proc. Cambridge Philos. Soc. 134 (2003), 1-12. doi: 10.1017/S0305004102006266 Zbl1053.11006
  5. [5] T. Komatsu, On Hurwitzian and Tasoev's continued fractions, Acta Arith. 107 (2003), 161-177. doi: 10.4064/aa107-2-4 Zbl1026.11012
  6. [6] T. Komatsu, Recurrence relations of the leaping convergents, JP J. Algebra Number Theory Appl. 3 (2003), 447-459. Zbl1178.11008
  7. [7] T. Komatsu, Arithmetical properties of the leaping convergents of e 1 / s , Tokyo J. Math. 27 (2004), 1-12. doi: 10.3836/tjm/1244208469 Zbl1075.11004
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  10. [10] T. Komatsu, An algorithm of infinite sums representations and Tasoev continued fractions, Math. Comp. 74 (2005), 2081-2094. doi: 10.1090/S0025-5718-05-01752-7 Zbl1074.11005
  11. [11] T. Komatsu, Some combinatorial properties of the leaping convergents, in: Combinatorial Number Theory, Proceedings of the Integers Conference 2005 in Celebration of the 70th Birthday of Ronald Graham, Carrollton, Georgia, USA, October 27-30,2005, eds. by B.M. Landman, M.B. Nathanson, J. Nesetril, R.J. Nowakowski and C. Pomerance, Walter de Gruyter, 2007, pp. 315-325. Zbl1178.11009
  12. [12] T. Komatsu, Some combinatorial properties of the leaping convergents, II, Applications of Fibonacci Numbers, Proceedings of 12th International Conference on Fibonacci Numbers and their Applications, Congr. Numer. 200 (2010), 187-196. Zbl1203.11010
  13. [13] T. Komatsu, Hurwitz continued fractions with confluent hypergeometric functions, Czech. Math. J. 57 (2007), 919-932. doi: 10.1007/s10587-007-0085-1 Zbl1163.11009
  14. [14] T. Komatsu, Leaping convergents of Hurwitz continued fractions, in: Diophantine Analysis and related fields (DARF 2007/2008), AIP Conf. Proc. 976, pp. 130-143. Amer. Inst. Phys., Melville, NY, 2008. Zbl1144.11007
  15. [15] T. Komatsu, Shrinking the period length of quasi-periodic continued fractions, J. Number Theory 129 (2009), 358-366. doi: 10.1016/j.jnt.2008.08.004 Zbl1219.11010
  16. [16] T. Komatsu, A diophantine appriximation of e 1 / s in terms of integrals, Tokyo J. Math. 32 (2009), 159-176. doi: 10.3836/tjm/1249648415 Zbl1241.11076
  17. [17] T. Komatsu, Diophantine approximations of tanh, tan, and linear forms of e in terms of integrals, Rev. Roum. Math. Pures Appl. 54 (2009), 223-242. Zbl1199.11098
  18. [18] B.G. Tasoev, Rational approximations to certain numbers (Russian), Mat. Zametki 67 (2000), 931-937; English transl. in Math. Notes 67 (2000), 786-791. 

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