Leaping convergents of Hurwitz continued fractions

Takao Komatsu

Discussiones Mathematicae - General Algebra and Applications (2011)

  • Volume: 31, Issue: 1, page 101-121
  • ISSN: 1509-9415

Abstract

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Let pₙ/qₙ = [a₀;a₁,...,aₙ] be the n-th convergent of the continued fraction expansion of [a₀;a₁,a₂,...]. Leaping convergents are those of every r-th convergent p r n + i / q r n + i (n = 0,1,2,...) for fixed integers r and i with r ≥ 2 and i = 0,1,...,r-1. The leaping convergents for the e-type Hurwitz continued fractions have been studied. In special, recurrence relations and explicit forms of such leaping convergents have been treated. In this paper, we consider recurrence relations and explicit forms of the leaping convergents for some different types of Hurwitz continued fractions.

How to cite

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Takao Komatsu. "Leaping convergents of Hurwitz continued fractions." Discussiones Mathematicae - General Algebra and Applications 31.1 (2011): 101-121. <http://eudml.org/doc/276703>.

@article{TakaoKomatsu2011,
abstract = {Let pₙ/qₙ = [a₀;a₁,...,aₙ] be the n-th convergent of the continued fraction expansion of [a₀;a₁,a₂,...]. Leaping convergents are those of every r-th convergent $p_\{rn+i\}/q_\{rn+i\}$ (n = 0,1,2,...) for fixed integers r and i with r ≥ 2 and i = 0,1,...,r-1. The leaping convergents for the e-type Hurwitz continued fractions have been studied. In special, recurrence relations and explicit forms of such leaping convergents have been treated. In this paper, we consider recurrence relations and explicit forms of the leaping convergents for some different types of Hurwitz continued fractions.},
author = {Takao Komatsu},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {Leaping convergents; Hurwitz continued fractions; leaping convergents},
language = {eng},
number = {1},
pages = {101-121},
title = {Leaping convergents of Hurwitz continued fractions},
url = {http://eudml.org/doc/276703},
volume = {31},
year = {2011},
}

TY - JOUR
AU - Takao Komatsu
TI - Leaping convergents of Hurwitz continued fractions
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2011
VL - 31
IS - 1
SP - 101
EP - 121
AB - Let pₙ/qₙ = [a₀;a₁,...,aₙ] be the n-th convergent of the continued fraction expansion of [a₀;a₁,a₂,...]. Leaping convergents are those of every r-th convergent $p_{rn+i}/q_{rn+i}$ (n = 0,1,2,...) for fixed integers r and i with r ≥ 2 and i = 0,1,...,r-1. The leaping convergents for the e-type Hurwitz continued fractions have been studied. In special, recurrence relations and explicit forms of such leaping convergents have been treated. In this paper, we consider recurrence relations and explicit forms of the leaping convergents for some different types of Hurwitz continued fractions.
LA - eng
KW - Leaping convergents; Hurwitz continued fractions; leaping convergents
UR - http://eudml.org/doc/276703
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 and T. Komatsu, A recurrence formula for leaping convergents of non-regular continued fractions, Linear Algebra Appl. 428 (2008), 824-833. doi: 10.1016/j.laa.2007.08.011 Zbl1132.05007
  3. [3] T. Komatsu, Recurrence relations of the leaping convergents, JP J. Algebra Number Theory Appl. 3 (2003), 447-459. Zbl1178.11008
  4. [4] 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
  5. [5] T. Komatsu, Some combinatorial properties of the leaping convergents, p. 315-325 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. 
  6. [6] 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
  7. [7] 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

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