Identities arising from binomial-like formulas involving divisors of numbers
Abstract
In this article, we derive a great number of identities involving the ω function counting distinct prime divisors of a given number n. These identities also include Pochhammer symbols, Fibonacci and Lucas numbers and many more.
Keywords
divisor; multiplicative function; symmetric polynomial; Fibonacci numbers; Stirling numbers
References
R. Battaloglu and Y. Simsek, On new formulas of Fibonacci and Lucas numbers involving golden ratio associated with atomic structure in chemistry, Symmetry 13 (2021), no. 8, Paper No. 1334, 10 pp.
K. Gryszka, Binomial formulas via divisors of numbers, Notes Number Theory Discrete Math. 27 (2021), no. 4, 122–128.
G.H. Hardy and E.M. Wright,An Introduction to the Theory of Numbers, sixth edition, Oxford University Press, Oxford, 2008.
R. Jakimczuk, On the function ω(n), Int. Math. Forum 13 (2018), no. 3, 107–116.
S. Lang, Algebra, revised third edition, Graduate Texts in Mathematics, 211, Springer-Verlag, New York, 2002.
G. Liu, An identity involving the Lucas numbers and Stirling numbers, Fibonacci Quart. 46/47 (2008/2009), no. 2, 136–139.
N.J.A. Sloane, The On-Line Encyclopedia of Integer Sequences, 2021. https://oeis.org/A105278
M.V. Vassilev-Missana, New form of the Newton’s binomial theorem, Notes Number Theory Discrete Math. 25 (2019), no. 1, 48–49.
T. Wakhare, Sums involving the number of distinct prime factors function, Rose-Hulman Undergrad. Math. J. 19 (2018), no. 1, Art. 8, 13 pp.
Instytut Matematyki, Uniwersytet Pedagoiczny im. Komisji Edukacji Narodowej w Krakowie Poland
https://orcid.org/0000-0002-3258-3330
This work is licensed under a Creative Commons Attribution 4.0 International License.
The Copyright Holders of the submitted text are the Author and the Journal. The Reader is granted the right to use the pdf documents under the provisions of the Creative Commons 4.0 International License: Attribution (CC BY). The user can copy and redistribute the material in any medium or format and remix, transform, and build upon the material for any purpose.
- License
This journal provides immediate open access to its content under the Creative Commons BY 4.0 license (http://creativecommons.org/licenses/by/4.0/). Authors who publish with this journal retain all copyrights and agree to the terms of the above-mentioned CC BY 4.0 license. - Author’s Warranties
The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s. - User Rights
Under the Creative Commons Attribution license, the users are free to share (copy, distribute and transmit the contribution) and adapt (remix, transform, and build upon the material) the article for any purpose, provided they attribute the contribution in the manner specified by the author or licensor. - Co-Authorship
If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.
