ϕ−Multiplicative Calculus

Year 2024, , 159 - 172, 31.07.2024

Orhan Dişkaya

https://doi.org/10.54974/fcmathsci.1412066

Abstract

In this paper, we present a novel mathematical framework termed “ϕ-multiplicative calculus”, which serves as a Golden Fibonacci calculus to fundamental concepts in multiplicative calculus. This innovative calculus introduces a parameter ϕ (Golden ratio), offering a nuanced extension of traditional calculus. Our work encompasses the establishment of ϕ-multiplicative calculus and the demonstration of essential theorems concerning derivatives, integrals, and their operation properties within this mathematical framework. The paper contributes to the academic discourse by providing a comprehensive exploration of the proposed ϕ-multiplicative calculus, presenting a robust foundation for further investigations in this specialized mathematical domain.

Keywords

Fibonacci numbers, multiplicative derivative, golden calculus, integral

References

• Bashirov A.E., Kurpınar E.M., Özyapıcı A., Multiplicative calculus and its applications, Journal of Mathematical Analysis and Applications, 337(1), 36-48, 2008.
• Bashirov A., Rıza M., On complex multiplicative differentiation, TWMS Journal of Applied and Engineering Mathematics, 1, 75-85, 2011.
• Bashirov A., On line and double multiplicative integrals, TWMS Journal of Applied and Engineering Mathematics, 3(1), 103-107, 2013.
• Boyer C.B., Merzbach U.C., A History of Mathematics, John Wiley & Sons, 2011.
• Calinger R.S., Leonhard Euler: Mathematical Genius in the Enlightenment, Princeton University Press, 2015.
• Florack L., Assen H.V., Multiplicative calculus in biomedical image analysis, Journal of Mathematical Imaging and Vision, 42, 64-75, 2012.
• Georgiev S.G., Zennir K., Multiplicative Differential Calculus, CRC Press, Boca Raton, 2023.
• Göktaş S., Yilmaz E., Yar A.C., Multiplicative derivative and its basic properties on time scales, Mathematical Methods in the Applied Sciences, 45(4), 2097-2109, 2022.
• Gurefe Y., Multiplicative Differential Equations and Its Applications, Master Thesis, Ege University, Institute of Sciences, İzmir, Türkiye, 2013.
• Grossman M., Katz R., Non-Newtonian Calculus, Lee Press, Pigeon Cove, 1972.
• Kac V.G., Cheung P., Quantum Calculus (Vol. 113), Springer, 2002.
• Koshy T., Fibonacci and Lucas Numbers with Applications (Vol. 1)„ John Wiley & Sons, 2018.
• Krot E., An introduction to finite fibonomial calculus, Open Mathematics, 2(5), 754-766, 2004.
• Kuş S., Tuğlu N., Kim T., Bernoulli F−polynomials and Fibo–Bernoulli matrices, Advances in Difference Equations, 145, 1-16, 2019.
• Misirli E., Gurefe Y., Multiplicative adams bashforth–moulton methods, Numerical Algorithms, 57, 425-439, 2011.
• Mora M., Córdova-Lepe F., Del-Valle R., A non-Newtonian gradient for contour detection in images with multiplicative noise, Pattern Recognition Letters, 33(10), 1245-1256, 2012.
• Özvatan M., Generalized Golden-Fibonacci Calculus and Applications, Master Thesis, İzmir Institute of Technology, Türkiye, 2018.
• Pashaev O.K., Nalci S., Golden quantum oscillator and Binet–Fibonacci calculus. Journal of Physics A: Mathematical and Theoretical, 45(1), 015303, 2011.
• Riza M., Özyapici A., Misirli E., Multiplicative finite difference methods, Quarterly of Applied Mathematics, 67(4), 745-754, 2009.
• Sadjang P.N., On the fundamental theorem of (p, q)−calculus and some (p, q)−Taylor formulas, Results in Mathematics, 73, 1-21, 2018.
• Stanley D., A multiplicative calculus, Problems, Resources, and Issues in Mathematics Undergraduate Studies, 9(4), 310-326, 1999.
• Yener G., Emiroğlu T., A q−analogue of the multiplicative calculus: q−multiplicative calculus, Discrete and Continuous Dynamical Systems, 8(6), 1435-1450, 2015.