On an expansion real numbers on some sequences
https://doi.org/10.22405/2226-8383-2022-23-3-50-60
Abstract
In this paper theorems on the expression of real numbers on multiplicative number system, Fibonacci sequence and integral valued sequences satisfiing recurrent correlations and connected
with Pisot–Vidgajraghavan, are proven. It pay a special attention to “explicit formulas” and conditions of the uniqueness of such representations. We note that unifiing of an expression of
a real number over inverse values of a multiplicaticative system permits to get the estimation of the form
$$𝑒 −Σ︁𝑛𝑘=0 1/𝑘!=𝑥𝑛/𝑛!, 1/𝑛 + 1≤ 𝑥𝑛 <1/𝑛.$$
Expressions of numbers over the sequence of inverse of Fibonacci numbers essentially uses these representation throw powers of “the gold section” 𝜙 = (1+√5)/2 .
Systems numbers connected with Pisot–Vidgajraghavana were considered less than in details, as demands to make a properties of examinated numbers more concrete.
About the Authors
Azar Giyasi
Allameh Tabataba’i University
Islamic Republic of Iran
Islamic Republic of Iran
candidate of physical and mathematical sciences
Ilya Petrovich Mikhailov
Kazan Aviation Institute
Russian Federation
Russian Federation
Vladimir Nikolaevich Chubarikov
doctor of physical and mathematical sciences, professor
Russian Federation
Russian Federation
Lomonosov Moscow State University
References
1. Hardy G. H., Littlewood J. E. (1914).The fractional part of 𝑛𝑘𝜃.// Acta math., 37.
2. Borel E. (1909). Les probabilit´es d´enombarables et leurs applications arithm´etiques.// Rend
3. Circolo math. Palermo, 27.
4. Gel’fond A. O. (1959). On one general property of numerical system// Izv. AN SSSR, Ser.
5. math. (in Russian). 23 (Selected works. p.366-371).
6. Zeckendorf E. (1972). Repr´sentation des nombres naturels par une somme de nombres de
7. Fibonacci ou de nombres de Lucas// Bull. Soc. R. Sci. Li`ege (in French). 41, p. 179-182.
8. Dickson L. E. (1919). History of the theory of numbers. — Carnegie Inst. of Washigton. Ch.17.
9. Arkhipov G. I., Sadovnichii V. A., Chubarikov V. N. (2006).Lectures on mathematical analysis.
10. — M.: Drofa. Pp. 640.
11. Cassels J. W. S. (1961). An introduction to Diophantine approximation. — Cambridge University
12. Press. Pp.212.
13. Hall M.,Jr. (1970). Combinatorial theory. — Waltham (Massachusetts)-Toronto-London: Blaisdell
14. Publ. Comp. Pp. 424.
15. Bernoulli D. (1728). Combinatorial theory.// Comment. Acad.Sci. Petrop., 3, p. 85–100.
16. Knuth D. E. (1998). The art computer programming. Fundamental algorithms. Third Ed. —
17. Reading, Massachusetts-Harlow, England-Menlo Park, California-Berkley, california-Lon Mills,
18. Ontario-Sidney-Bonn-Amsterdam-Tokyo-Mexico City: AddisonWesley Longman, Inc.. Pp. 720.
19. de Moivre A. (1922). // Philos. Trans., 32, p. 162–178.
20. ChebyshevP. L. (1936). The theory of probabilities. — AN SSSR. S23. 143–147. (in Russian).
21. Landau E. (1947). Fundamentals of analysis. — M.: Inostr.literature.(in Russian).
22. Golubov B. I., Efimov A. V., Skvortsov V. A. (1987). Series and the Uolsh’s transformations: the
23. theory and applications. — M.: Nauka, pp. 344.(in Russian).
24. Mineev M.P., Chubarikov V. N. (2014). Lectures on arithmetical questions of cryptography. —
25. M.: OOO“Luch”, pp. 224. (in Russian).
26. Ghyasi А. H. (2007). A generalization of the Gel’fond theorem concerning number systems//
27. Russian Journal of Mathematical Physics. 14, No.3, p.370.
Review
For citations:
Giyasi A., Mikhailov I.P., Chubarikov V.N. On an expansion real numbers on some sequences. Chebyshevskii Sbornik. 2022;23(3):50-60. (In Russ.) https://doi.org/10.22405/2226-8383-2022-23-3-50-60
Views:
297
Email this article 