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Consecutive Ratios in Second-Order Linear Recurrence Sequences Cover

Consecutive Ratios in Second-Order Linear Recurrence Sequences

By: Daniel Berend and  Rishi Kumar  
Open Access
|Dec 2022

References

  1. [1] ABUZAID, A. H.—EL-HANJOURI, M. M.—KULAB, M. M.: On discordance tests for the wrapped Cauchy distribution,Open J.Statist. 5 (2015), no. 4, 245–253.
  2. [2] BAGDASAR, O.—HEDDERWICK, E.—POPA, I. L.: On the ratios and geometric boundaries of complex Horadam sequences, Electron. Notes Discrete Math. 67 (2018), 63–70.10.1016/j.endm.2018.05.011
  3. [3] BEARDON, A. F.: Iteration of Rational Functions. Springer-Verlag, New York, 1991.10.1007/978-1-4612-4422-6
  4. [4] CARATHÉODORY, C.: Theory of Functions of a Complex Variable. Vol. 1., Vol. 2. Translated by F. Steinhardt. Chelsea Publishing Co., New York, N. Y. 1954.
  5. [5] DOWNS, T. D.: Cauchy families of directional distributions closed under location and scale transformations, Open Stat. Prob. J. 1 (2009), no. 1, 76–92.
  6. [6] DOWNS,T.D.—DOWNS,K. J.: Linear and directional domains with Cauchy probability distributions,Open Stat. Prob. J. 4 (2012), no. 1, 5–10.
  7. [7] FIORENZA, A. —VINCENZI, G.: Limit of ratio of consecutive terms for general order-k linear homogeneous recurrences with constant coefficients, Chaos Solitons Fractals 44 (2011), no. 1–3, 145–152.
  8. [8] GOLDSTERN, M.—TICHY, R. F.—TURNWALD, G.: Distribution of the ratios of the terms of a linear recurrence, Monatsh. Math. 107 (1989), no. 1, 35–55.
  9. [9] GOLZY, M.—MARKATOU, M.: Poisson kernel-based clustering on the sphere: convergence properties, identifiability, and a method of sampling, J. Comput. Graph. Statist. 29 (2020), no. 4 758–770.
  10. [10] JAMMALAMADAKA, S. R.—SENGUPTA, A.: Topics in Circular Statistics. Series on Multivariate Analysis, Vol 5. World Scientific Publishing Co., Inc., River Edge, NJ, 2001.10.1142/4031
  11. [11] KATO, S.—MCCULLAGH, P.: Some properties of a Cauchy family on the sphere derived from the Möbius transformations, Bernoulli 26 (2020), no. 4, 3224–3248.
  12. [12] KATO, S.—SHIMIZU, K.—SHIEH, G.: A circular-circular regression model, Statist. Sinica 18 (2008), no. 2, 633–645.
  13. [13] KATOK, A.—HASSELBLATT, B.: Introduction to the Modern Theory of Dynamical Systems. With a supplementary chapter by Katok and Leonardo Mendoza. Encyclopedia of Mathematics and its Applications, Vol. 54. Cambridge University Press, Cambridge, 1995.
  14. [14] KELLEY, W. G.—PETERSON, A. C.: Difference Equations: An Introduction with Applications. Second edition. Harcourt/Academic Press, San Diego, California, 2001.
  15. [15] KISS, P.: A Diophantine approximative property of the second-order linear recurrences, Period. Math. Hungar. 11 (1980), no. 4, 281–287.
  16. [16] KISS, P.: Results on the ratios of the terms of second-order linear recurrences,Mathematica Slovaca 41 (1991), no. 3, 257-260.
  17. [17] KISS, P.: An approximation problem concerning linear recurrences,In: Number Theory (Eger, 1996), De Gruyter, Berlin, 1998, pp. 289–293.
  18. [18] KISS, P.—SINKA, Z.: On the ratios of the terms of second-order linear recurrences, Period. Math. Hungar. 23 (1991), no. 2, 139–143.
  19. [19] KISS, P.—TICHY, R. F.: Distribution of the ratios of the terms of a second-order linear recurrence, Nederl. Akad. Wetensch. Indag. Math. 48 (1986), no. 1, 79–86.
  20. [20] KISS, P.—TICHY, R. F.: A discrepancy problem with applications to linear recurrences I, Proc. Japan Acad. Ser. A Math. Sci. 65 (1989), no. 5, 135–138.
  21. [21] KISS, P.—TICHY, R. F.: A discrepancy problem with applications to linear recurrences II, Proc. Japan Acad. Ser. A Math. Sci., 65 (1989), no. 6, 191–194.
  22. [22] KOSHY, T.: Fibonacci and Lucas Numbers with Applications. John Wiley & Sons, New York, 2001.10.1002/9781118033067
  23. [23] KUIPERS, L.—NIEDERREITER, H.: Uniform Distribution of Sequences. John Wiley, New York, 1974.
  24. [24] LÉVY, P.: L’addition des variables aléatoires définies sur une circonférence, Bull. Soc. Math. France 67 (1939), 1–41.
  25. [25] MARDIA, K. V.: Statistics of Directional Data, Academic Press, New York, 1972.
  26. [26] MCCULLAGH, P.: Möbius transformation and Cauchy parameter estimation, Ann. Statist. 24 (1996), no. 2, 787–808.
  27. [27] POINCARÉ, H.: Sur leséquations linéaires aux différentielles ordinaires et aux différences finies,Amer. J. Math. 7 (1885), no. 3, 203–258.
  28. [28] RAVINDRAN, P.—GHOSH, S.: Bayesian analysis of circular data using wrapped distributions, J. Stat. Theory Pract. 5 (2011), no. 4, 547–561.
  29. [29] ROSS, S. M.: A First Course in Probability. Pearson Education Limited, Upper Saddle River, New Jersey, 2010.
  30. [30] RUDIN, W.: Real and Complex Analysis. McGraw-HiII, New York, 1987.
  31. [31] VIANA, W.—OLIVEIRA, K.: Foundations of Ergodic Theory. Cambridge University Press, Cambridge, 2016.10.1017/CBO9781316422601
  32. [32] WALCK, C.: Hand-book on Statistical Distributions for Experimentalists. Internal Report SUF-PFY/96-01 (last modification 10 Sept. 2007), Fysikum, University of Stockholm, Particle Physics Group, 2007.
  33. [33] WALTERS, P.: An Introduction to Ergodic Theory. Springer-Verlag, Berlin, 1982.10.1007/978-1-4612-5775-2
  34. [34] WINTNER, A.: On the shape of the angular case of Cauchy’s distribution curves, Annals Math. Statist. 18 (1947), no. 4, 589–593.
DOI: https://doi.org/10.2478/udt-2022-0012 | Journal eISSN: 2309-5377 | Journal ISSN: 1336-913X
Language: English
Page range: 51 - 76
Submitted on: Feb 22, 2022
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Accepted on: Apr 21, 2022
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Published on: Dec 12, 2022
In partnership with: Paradigm Publishing Services
Publication frequency: 2 issues per year

© 2022 Daniel Berend, Rishi Kumar, published by Slovak Academy of Sciences, Mathematical Institute
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.