Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
권호 표지

SOME REMARKS ON THE PERIODIC CONTINUED FRACTION

  • Lee, Yeo-Rin (Department of Mathematics Eduacation Pusan University of Foreign Studies)
  • Received : 2009.01.11
  • Accepted : 2009.02.17
  • Published : 2009.06.30
Download PDF
⟨ Previous Next ⟩

Abstract

Using the Binet's formula, we show that the quotient related ratio $l_{1(x)}\;\neq\;0$ for the eventually periodic continued fraction x. Using this ratio, we also show that the derivative of the Minkowski question mark function at the simple periodic continued fraction is infinite or 0. In particular, $l_1({[\bar{1}]})$ = 2 log $\gamma$ where $\gamma$ is the golden mean $(1+\sqrt{5})/2$ and the derivative of the Minkowski question mark function at the simple periodic continued fraction $[\bar{1}]$ is infinite.

Keywords

References

  1. I. S. Baek, A note on the moments of the Riesz-Nagy-Takacs distribution, Journal of Mathematical Analysis and Applications 348 (2008), no. 1, 165-168. https://doi.org/10.1016/j.jmaa.2008.07.014
  2. M. Kessebohmer and B. O. Stratmann, A multifractal analysis for Stern-Brocot intervals, continued fractions and Diophantine growth rates, Journal fur die reine und angewandte Mathematik 605 (2007), 133-163.
  3. M. Kessebohmer and B. O. Stratmann, Fractal analysis for sets of nondifferentiability of Minkowski's question mark function, Journal of Number Theory 128 (2008), 2663-2686. https://doi.org/10.1016/j.jnt.2007.12.010
  4. M. Livio, The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books (2002).
  5. J. Paradis, P. Viader, L. Bibiloni, The Derivative of Minkowski's -(x) Function, Journal of Mathematical Analysis and Applications 253 (2001), no. 1, 107-125. https://doi.org/10.1006/jmaa.2000.7064