An operational calculus model for the central difference and exponential-trigonometric and hyperbolic fibonacci sequences

Wysocki, Hubert

doi:10.2478/sjpna-2018-0018

Abstract

In the paper, there has been constructed such a non-classical Bittner operational calculus model, in which the derivative is understood as a central difference D_n{x(k)}:= {x(k+n)–x(k-n)}. The discussed model has been generalized by considering the operation D_n,b{x(k)}:= {x(k+n)–bx(k-n)}, where b∈𝕔\{0}. In the D₁-difference model exponential-trigonometric and hyperbolic Fibonacci sequences have been introduced.

References

[1] Apostol T. M., Calculus, Vol. 1, One-Variable Calculus, with an Introduction to Linear Algebra, John Wiley & Sons, New York — London 1967.
Search in Google Scholar Back to article
[2] Bittner R., On certain axiomatics for the operational calculus, ‘Bull. Acad. Polon. Sci.’, Cl. III, 1959, 7(1), pp. 1–9.
Search in Google Scholar Back to article
[3] Bittner R., Operational calculus in linear spaces, ‘Studia Math.’, 1961, 20, pp. 1–18.10.4064/sm-20-1-1-18
Search in Google Scholar Back to article
[4] Bittner R., Algebraic and analytic properties of solutions of abstract differential equations, ‘Rozprawy Matematyczne’ [‘Dissertationes Math.’], 42, PWN, Warszawa 1964.
Search in Google Scholar Back to article
[5] Bittner R., Rachunek operatorów w przestrzeniach liniowych, PWN, Warszawa 1974 [Operational Calculus in Linear Spaces — available in Polish].
Search in Google Scholar Back to article
[6] Bittner R., Mieloszyk E., About eigenvalues of differential equations in the operational calculus, ‘Zeszyty Naukowe Politechniki Gdańskiej, Matematyka XI’, 1978, 285, pp. 87–99.
Search in Google Scholar Back to article
[7] Elaydi S., An Introduction to Difference Equations, Springer Sci. & Business Media, New York 2005.
Search in Google Scholar Back to article
[8] Falcón S., Plaza A., On the Fibonacci k-numbers, ‘Chaos, Solitons and Fractals’, 2007, 32(5), pp. 1615–1624.10.1016/j.chaos.2006.09.022
Search in Google Scholar Back to article
[9] Gazalé M. J., Gnomon: From Pharaohs to Fractals, Princeton Univ. Press, New Jersey 1999.
Search in Google Scholar Back to article
[10] Jordan Ch., Calculus of Finite Differences, Chelsea Publ. Comp., New York 1950.
Search in Google Scholar Back to article
[11] Kalman D., Mena R., The Fibonacci numbers — exposed, ‘Math. Magazine’, 2003, 76(3), pp. 167–181.10.1080/0025570X.2003.11953176
Search in Google Scholar Back to article
[12] Levy H., Lessman F., Finite Difference Equations, Pitman & Sons, London 1959.
Search in Google Scholar Back to article
[13] Mathews J. H., Fink K. D., Numerical Methods Using MATLAB, Prentice Hall, New Jersey 1999.
Search in Google Scholar Back to article
[14] Mercer P. R., More Calculus for a Single Variable, Springer Sci. & Business Media, New York 2014.10.1007/978-1-4939-1926-0
Search in Google Scholar Back to article
[15] Mikusiński J., Operational Calculus, Pergamon Press, London 1959.
Search in Google Scholar Back to article
[16] Spinadel V. W., The Family of Metallic Means, ‘VisMath — Visual Mathematics’, Electronic Journal, 1999, Vol. 1(3) [online], http://www.mi.sanu.ac.rs/vismath/spinadel/index.html [access 14.06.2018].
Search in Google Scholar Back to article
[17] Spinadel V. W., New Smarandache Sequences: The Family of Metallic Means [online], http://vixra.org/abs/1403.0507 [access 14.06.2018].
Search in Google Scholar Back to article
[18] Stakhov A., The Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science, Series of Knots and Everything: Vol. 22, World Scientific, Singapore 2009.10.1142/6635
Search in Google Scholar Back to article
[19] Stakhov A., Rozin B., On a new class of hyperbolic functions, ‘Chaos, Solitons and Fractals’, 2005, 23(2), pp. 379–389.10.1016/j.chaos.2004.04.022
Open DOI Search in Google Scholar Back to article
[20] Washburn L., The Lanczos derivative, Senior Project Archive 2006, Dept. of Maths., Whitman College, USA, [online], https://www.whitman.edu/Documents/Academics/Mathematics/washbuea.pdf [access 14.06.2018].
Search in Google Scholar Back to article
[21] Wysocki H., Spira Mirabilis in the selected models of the Bittner operational calculus, ‘Zeszyty Naukowe Akademii Marynarki Wojennej’ [‘Scientific Journal of PNA’], 2015, 4(203), pp. 65–96.10.5604/0860889X.1187627
Search in Google Scholar Back to article
[22] Wysocki H., An operational calculus model for the n^th-order forward difference, ‘Zeszyty Naukowe Akademii Marynarki Wojennej’ [‘Scientific Journal of PNA’], 2017, 3(210), pp. 107–117.10.5604/01.3001.0010.6593
Search in Google Scholar Back to article

An operational calculus model for the central difference and exponential-trigonometric and hyperbolic fibonacci sequences

Abstract

Paradigm

My account