Leaping convergents of Tasoev continued fractions

Komatsu, Takao

Discussiones Mathematicae. General Algebra and Applications, Tome 31 (2011) no. 2, pp. 201-216.

Résumé

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.

Détail

Keywords: leaping convergents, Tasoev continued fractions

Bibliographie

[1] C. Elsner, On arithmetic properties of the convergents of Euler's number, Colloq. Math. 79 (1999), 133-145.

[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

[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

[4] T. Komatsu, On Tasoev's continued fractions, Math. Proc. Cambridge Philos. Soc. 134 (2003), 1-12. doi: 10.1017/S0305004102006266

[5] T. Komatsu, On Hurwitzian and Tasoev's continued fractions, Acta Arith. 107 (2003), 161-177. doi: 10.4064/aa107-2-4

[6] T. Komatsu, Recurrence relations of the leaping convergents, JP J. Algebra Number Theory Appl. 3 (2003), 447-459.

[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

[8] T. Komatsu, Tasoev's continued fractions and Rogers-Ramanujan continued fractions, J. Number Theory 109 (2004), 27-40. doi: 10.1016/j.jnt.2004.06.001

[9] T. Komatsu, Hurwitz and Tasoev continued fractions, Monatsh. Math. 145 (2005), 47-60. doi: 10.1007/s00605-004-0281-0

[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

[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.

[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.

[13] T. Komatsu, Hurwitz continued fractions with confluent hypergeometric functions, Czech. Math. J. 57 (2007), 919-932. doi: 10.1007/s10587-007-0085-1

[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.

[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

[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

[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.

[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.