Composition Operations of Generalized Entropies Applied to the Study of Numbers

Author(s): Amelia Carolina Sparavigna

The generalized entropies of C. Tsallis and G. Kaniadakis have composition operations, which can be applied to the study of numbers. Here we will discuss these composition rules and use them to study some famous sequences of numbers (Mersenne, Fermat, Cullen, Woodall and Thabit numbers). We will also consider the sequence of the repunits, which can be seen as a specific case of q-integers.

Generalized Entropies, Q-Calculus, Abelian Groups, Hyperbolic Functions, Fermat Numbers, Mersenne Numbers, Thabit Numbers, Repunits

1. Sparavigna, A. C. (2015). Shannon, Tsallis and Kaniadakis entropies in bi-level image thresholding. International Journal of Sciences 4(2), 35-43. DOI: 10.18483/ijSci.626
2. Sparavigna, A. C. (2015). Relations between Tsallis and Kaniadakis entropic measures and rigorous discussion of conditional Kaniadakis entropy. International Journal of Sciences, 4(10), 47-50. DOI: 10.18483/ijsci.866
3. Sparavigna, A. C. (2015), On The Generalized Additivity Of Kaniadakis Entropy, International Journal of Sciences 4(2), 44-48. DOI: 10.18483/ijSci.627
4. Tsallis, C. (1988). Possible generalization of Boltzmann-Gibbs statistics. Journal of Statistical Physics, 52(1-2), 479-487. DOI: 10.1007/bf01016429
5. Kaniadakis, G. (2001). Non-linear kinetics underlying generalized statistics. Physica A, 296, 405–425. DOI: 10.1016/s0378-4371(01)00184-4
6. Kaniadakis, G. (2013). Theoretical Foundations and Mathematical Formalism of the Power-Law Tailed Statistical Distributions. Entropy, 15, 3983-4010. DOI: 10.3390/e15103983
7. Sparavigna, A. C. (2015). Gray-level image transitions driven by Tsallis entropic index. International Journal of Sciences 4(2), 16-25. DOI: 10.18483/ijSci.621
8. Portes De Albuquerque, M., Esquef, I. A., Gesualdi Mello, A. R., & Portes De Albuquerque, M. (2004). Image thresholding using Tsallis entropy. Pattern Recognition Letters, 25(9), 1059-1065. DOI: 10.1016/j.patrec.2004.03.003
9. Sparavigna, A. C. (2018). Generalized Sums Based on Transcendental Functions. SSRN Electronic Journal. Elsevier. DOI: 10.2139/ssrn.3171628
10. Scarfone, A. M. (2013). Entropic forms and related algebras. Entropy, 15(2), 624-649. DOI: 10.3390/e15020624
11. Sicuro, G., & Tempesta, P. (2016). Groups, information theory, and Einstein's likelihood principle. Phys. Rev. E 93, 040101(R). DOI: 10.1103/physreve.93.040101
12. Tempesta, P. (2015). Groups, generalized entropies and L-series. Templeton Workshop on Foundations of Complexity, October 2015. http://www.cbpf.br/~complex/Files/talk_tempesta.pdf
13. Abramowitz, M., & Stegun, I. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables.
14. Kaniadakis, G. (2005). Statistical mechanics in the context of special relativity II. Phys. Rev. E, 72, 036108. DOI: 10.1103/physreve.72.036108
15. Ernst, T. (2012). A Comprehensive Treatment of q-Calculus, Springer Science & Business Media.
16. Annaby, M. H., & Mansour, Z. S. (2012). q-Fractional Calculus and Equations, Springer.
17. Ernst, T. (2000). The History of q-calculus and a New Method. Department of Mathematics, Uppsala University.
18. Aral, A., Gupta, V., & Agarwal, R. P. (2013). Applications of q-Calculus in Operator Theory. Springer Science & Business Media.
19. Ernst, T. (2008). The different tongues of q-calculus. Proceedings of the Estonian Academy of Sciences, 2008, 57, 2, 81–99 DOI: 10.3176/proc.2008.2.03
20. Kac, V., & Pokman Cheung (2002). Quantum Calculus, Springer, Berlin.
21. Sparavigna, A. C. (2018). The q-integers and the Mersenne numbers. SSRN Electronic Journal. Elsevier. DOI: 10.2139/ssrn.3183800
22. Sparavigna, A. C. (2018). On the generalized sum of the symmetric q-integers. Zenodo. DOI: 10.5281/zenodo.1248959
23. Weisstein, Eric W. “Mersenne Number”. From MathWorld - A Wolfram Web Resource. http://mathworld.wolfram.com/MersenneNumber.html
24. Weisstein, Eric W. “Fermat Number”. From MathWorld - A Wolfram Web Resource. http://mathworld.wolfram.com/FermatNumber.html
25. Sparavigna, A. C. (2018). The group of the Fermat Numbers. Zenodo. DOI: 10.5281/zenodo.1252422
26. Weisstein, Eric W. “Cullen Number”. From MathWorld - A Wolfram Web Resource. http://mathworld.wolfram.com/CullenNumber.html
27. Weisstein, Eric W. “Woodall Number”. From MathWorld - A Wolfram Web Resource. http://mathworld.wolfram.com/WoodallNumber.html
28. Sparavigna, A. C. (2019). On the generalized sums of Mersenne, Fermat, Cullen and Woodall Numbers. Zenodo. DOI: 10.5281/zenodo.2634312
29. Weisstein, Eric W. (2019). “Thâbit ibn Kurrah Number”. From MathWorld - A Wolfram Web Resource. http://mathworld.wolfram.com/ThabitibnKurrahNumber.html
30. Sparavigna, A. C. (2019). A recursive formula for Thabit numbers. Zenodo. DOI: 10.5281/zenodo.2638790
31. Weisstein, Eric W. (2019). “Repunit”. From MathWorld - A Wolfram Web Resource. http://mathworld.wolfram.com/Repunit.html
32. Beiler, A. H. (1966). "11111...111." Ch. 11 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. New York: Dover.
33. Sparavigna, A. C. (2019). On Repunits. DOI: 10.5281/zenodo.2639620