پیوندهای مفید
آمار
| تعداد نشریات | 22 |
| تعداد شمارهها | 717 |
| تعداد مقالات | 10,309 |
| تعداد مشاهده مقاله | 72,281,052 |
| تعداد دریافت فایل اصل مقاله | 64,010,089 |
On the distribution modulo 1 of the sequence $w(n) e n!$ | ||
| International Journal of Nonlinear Analysis and Applications | ||
| مقالات آماده انتشار، اصلاح شده برای چاپ، انتشار آنلاین از تاریخ 02 مرداد 1405 اصل مقاله (374.09 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22075/ijnaa.2025.36265.5354 | ||
| نویسندگان | ||
| Mohammad Ghorbani1؛ Mehdi Hassani* 1؛ Hossein Moshtagh2 | ||
| 1Department of Mathematics, Faculty of Science, University of Zanjan, University Blvd., 45371-38791, Zanjan, Iran | ||
| 2Department of Computer science, University of Garmsar, 35881-15589, Garmsar, Semnan, Iran | ||
| تاریخ دریافت: 25 آذر 1403، تاریخ بازنگری: 19 مرداد 1404، تاریخ پذیرش: 20 مرداد 1404 | ||
| چکیده | ||
| In this paper, we study the distribution modulo 1 of sequences of the form $a_n=w(n)\lbrace\mathrm{e} n!\rbrace$. We show that if the weight function $w$ is a nonlinear polynomial with a leading irrational coefficient, the $a_n$ is uniformly distributed modulo 1. Also, we show that if $w(n)=\alpha n+\beta$ (linear polynomial), then $w(n)\lbrace\mathrm{e} n!\rbrace\to\alpha$ as $n\to\infty$, and if $w(n)=O(n^{1-\epsilon})$ (sublinear growth) for some $\epsilon>0$, then $w(n)\lbrace\mathrm{e} n!\rbrace\to 0$ as $n\to\infty$. | ||
| کلیدواژهها | ||
| fractional part؛ distribution modulo 1؛ asymptotic expansion | ||
| مراجع | ||
|
[1] R.K. Akhunzhanov, On the distribution modulo 1 of exponential sequences (Russian), Math. Notes 76 (2004), no. 1-2, 153-160. [2] D. Berend, M.D. Boshernitzan, and G. Kolesnik, Irrational dilations of Pascal's triangle, Mathematika 48 (2001), no. 1-2, 159-168. [3] F. Brafman, On Touchard polynomials, Can. J. Math. 9 (1957), 191-193. [4] L. Carlitz, Some polynomials of Touchard connected with the Bernoulli numbers, Can. J. Math. 9 (1957), 188-190. [5] O. Chrysaphiou, On Touchard polynomials, Discrete Math. 54 (1985), 143-152. [6] N.G. de Bruijn, Asymptotic Methods in Analysis, North‑Holland Publishing Co., Amsterdam, 1961. [7] B. de Mahan, Numbers contravening a condition in density modulo 1, Acta Math. Acad. Sci. Hungar. 36 (1980), no. 3-4, 237-241. [8] P. Diaconis, The distribution of leading digits and uniform distribution mod 1, Ann. Probab. 5 (1977), no. 1, 72-81. [9] A. Dubickas, On the limit points of (anξ)n=1∞ mod 1 for slowly increasing integer sequences (an)n=1∞, Proc. Amer. Math. Soc. 137 (2009), no. 2, 449-456. [10] P. Erdős and S.J. Taylor, On the set of points of convergence of a lacunary trigonometric series and the equidistribution properties of related sequences, Proc. London Math. Soc. 7 (1957), no. 3, 598-615. [11] M. Hassani, Cycles in graphs and derangements, Math. Gaz. 88 (2004), 123-126. [12] M. Hassani, Enumeration by e, Modern Discrete Mathematics and Analysis: With Applications in Cryptography, Information Systems and Modelling, Themistocles M. Rassias and Nicholas J. Daras (Eds.), Springer, Switzerland, 2018, pp. 227-233. [13] M. Hassani, On a difference concerning the number e and summation identities of permutations, J. Inequal. Spec. Funct. 12 (2021), 14-22. [14] H. Iwaniec and E. Kowalski, Analytic Number Theory, AMS, 2004. [15] L. Kuipers and H. Niederreiter, Uniform distribution of Sequences, Dover Publication Inc., Mineola, New York, 2006. [16] O.V. Kuzmin and O.V. Leonova, Touchard polynomials and their applications, Discrete Math. Appl. 10 (2000), 391-402. [17] R.B. Paris, The asymptotics of the Touchard polynomials, Math. Aeterna 6 (2016), 765-779. [18] R.G. Pinsky, Some connections between permutation cycles and Touchard polynomials and between permutations that fix a set and covers of multisets, Electron. Commun. Probab. 22 (2017), Paper No. 17, 9 p. [19] A.D. Pollington, On the density of sequence {nkξ}, Illinois J. Math. 23 (1979), no. 4, 511-515. [20] W. Rudin, Principles of Mathematical Analysis, Third Edition, McGraw‑Hill, 1976. [21] N.J.A. Sloane, The on‑line encyclopedia of integer sequences, https://oeis.org. [22] W.M. Schmidt, On badly approximable numbers, Mathematika 12 (1965), 10-20. [23] W.M. Schmidt, On badly approximable numbers and certain games, Trans. Amer. Math. Soc. 123 (1966), 178-199. [24] E. Strzelecki, On sequences {ξtn(mod1)}, Canad. Math. Bull. 18 (1975), no. 5, 727-738. [25] J. Touchard, Sur les cycles des substitutions, Acta Math. 70 (1939), 243-297. [26] E.W. Weisstein, CRC Concise Encyclopedia of Mathematics (Second Edition), CRC Press, 2003. [27] H. Weyl, Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916), 313-352. [28] M. Wyman and L. Moser, On some polynomials of Touchard, Canad. J. Math. 8 (1956), 321-322. | ||
|
آمار تعداد مشاهده مقاله: 6 تعداد دریافت فایل اصل مقاله: 4 |
||