Open Access

The Generalized Shifts and Rational Numbers


Cite

[1] CANTOR, G.: Über die einfachen Zahlensysteme, Zeit für. Math. Phys. 14 (1869), 121–128. Search in Google Scholar

[2] DIANANDA, P. H.—OPPENHEIM, A.: Criteria for irrationality of certain classes of numbers II, Amer. Math. Monthly 62 (1955), no. 4, 222–225.10.1080/00029890.1955.11988618 Search in Google Scholar

[3] ERDŐS, P.—STRAUS, E. G.: On the irrationality of certain series, Pacific J. Math. 55 (1974), no. 1, 85–92.10.2140/pjm.1974.55.85 Search in Google Scholar

[4] GALAMBOS, J.: Representations of Real Numbers by Infinite Series. Lecture Notes in Math. Vol. 502, Springer-Verlag, Berlin, 1976.10.1007/BFb0081642 Search in Google Scholar

[5] HANČL, J.: A note to the rationality of infinite series I, Acta Math. Inform. Univ. Ostraviensis 5 (1997), no. 1, 5–11. Search in Google Scholar

[6] HANČL, J.—RUCKI, P.: A note to the transcendence of special infinite series, Math. Slovaca 56 (2006), no. 4, 409–414. Search in Google Scholar

[7] HANČL, J.—TIJDEMAN, R.: On the irrationality of Cantor series, J. Reine Angew. Math. 571 (2004), 145–158.10.1515/crll.2004.038 Search in Google Scholar

[8] KUHAPATANAKUL, P.—LAOHAKOSOL, V.: Irrationality of some series with rational terms, Kasetsart J. (Nat. Sci.) 35 (2001), 205–209. Search in Google Scholar

[9] OPPENHEIM, A.: Criteria for irrationality of certain classes of numbers, Amer. Math. Monthly 61 (1954), no. 4, 235–241.10.1080/00029890.1954.11988450 Search in Google Scholar

[10] SERBENYUK, S.: On some generalizations of real numbers representations, arXiv preprint arXiv:1602.07929 (2016). (In Ukrainian) Search in Google Scholar

[11] SERBENYUK, S.: Cantor series and rational numbers, arXiv preprint arXiv:1702.00471 (2017), available at https://arxiv.org/pdf/1702.00471.pdf Search in Google Scholar

[12] SERBENYUK, S.: Cantor series expansions of rational numbers, arXiv preprint arXiv:1706.03124 (2017), available at https://www.researchgate.net/publication/317099134 Search in Google Scholar

[13] SERBENYUK, S.: Representation of real numbers by the alternating Cantor series, Integers, 17 (2017), paper no. A15, 27 pp. Search in Google Scholar

[14] SERBENYUK, S.: A note on expansions of rational numbers by certain series, Tatra Mt. Math. Publ. 77 (2020), 53–58, doi: 10.2478/tmmp-2020-0032, arXiv:1904.07264. Open DOISearch in Google Scholar

[15] SERBENYUK, S.: Rational numbers defined in terms of certain generalized series, Acta Math. Hungar. 164 (2021), 580–592, https://doi.org/10.1007/s10474-021-01163-510.1007/s10474-021-01163-5 Search in Google Scholar

[16] SERBENYUK, S.: Systems of functional equations and generalizations of certain functions, Aequationes Math. 95 (2021), no. 5, 801–820, https://doi.org/10.1007/s00010-021-00840-810.1007/s00010-021-00840-8 Search in Google Scholar

[17] TIJDEMAN, R.—YUAN, P.: On the rationality of Cantor and Ahmes series, Indag. Math. (N.S.) 13 (2002), no. 3, 407–418.10.1016/S0019-3577(02)80018-0 Search in Google Scholar

eISSN:
1338-9750
Language:
English
Publication timeframe:
3 times per year
Journal Subjects:
Mathematics, General Mathematics