Multi Parameters Golden Ratio and Some Applications

ABSTRACT

The present paper is devoted to the generalized multi parameters golden ratio. Variety of features like two-dimensional continued fractions, and conjectures on geometrical properties concerning to this subject are also presented. Wider generalization of Binet, Pell and Gazale formulas and wider generalizations of symmetric hyperbolic Fibonacci and Lucas functions presented by Stakhov and Rozin are also achieved. Geometrical applications such as applications in angle trisection and easy drawing of every regular polygons are developed. As a special case, some famous identities like Cassini’s, Askey’s are derived and presented, and also a new class of multi parameters hyperbolic functions and their properties are introduced, finally a generalized Q-matrix called G_{n}-matrix of order *n* being a generating matrix for the generalized Fibonacci numbers of order n and its inverse are created. The corresponding code matrix will prevent the attack to the data based on previous matrix.

The present paper is devoted to the generalized multi parameters golden ratio. Variety of features like two-dimensional continued fractions, and conjectures on geometrical properties concerning to this subject are also presented. Wider generalization of Binet, Pell and Gazale formulas and wider generalizations of symmetric hyperbolic Fibonacci and Lucas functions presented by Stakhov and Rozin are also achieved. Geometrical applications such as applications in angle trisection and easy drawing of every regular polygons are developed. As a special case, some famous identities like Cassini’s, Askey’s are derived and presented, and also a new class of multi parameters hyperbolic functions and their properties are introduced, finally a generalized Q-matrix called G

Cite this paper

nullS. Hashemiparast and O. Hashemiparast, "Multi Parameters Golden Ratio and Some Applications,"*Applied Mathematics*, Vol. 2 No. 7, 2011, pp. 808-815. doi: 10.4236/am.2011.27108.

nullS. Hashemiparast and O. Hashemiparast, "Multi Parameters Golden Ratio and Some Applications,"

References

[1] D. H. Fowler, “A Generalization of the Golden Section,” Fibonacci Quarterly, Vol. 20, 1982, pp. 146-158.

[2] A. P. Stakhov, “The Generalized Principle of the Golden Section and Its Applications in Mathematics, Science and Engineering,” Chaos, Solitons and Fractals, Vol. 26, No. 2, 2005, pp. 263-289. doi:10.1016/j.chaos.2005.01.038

[3] A. Stakhov, “Fundamentals of a New Kind of Mathematics Based on the Golden Section,” Chaos, Solitons and Fractals, Vol. 27, No. 5, 2006, pp. 1124-1146. doi:10.1016/j.chaos.2005.05.008

[4] M. Basu and B. Prasad, “Coding Theory on the M-Extension of the Fibonacci P-Numbers,” Chaos, Solitons and Fractals, Vol. 42, No. 4, 2009, pp. 2522-2530. doi:10.1016/j.chaos.2009.03.197

[5] M. Basu and B. Prasad, “The Generalized Relations among the Code Elements for Fibonacci Coding Theory,” Chaos, Solitons and Fractals, Vol. 41, No. 5, 2009, pp. 2517-2525. doi:10.1016/j.chaos.2008.09.030

[6] E. G. Kocer, N. Tuglu and A. Stakhov, “On the M-Extension of the Fibonacci and Lucas P-Numbers, Chaos,” Chaos, Solitons and Fractals, Vol. 40, No. 4, 2009, pp. 1890-1906. doi:10.1016/j.chaos.2007.09.071

[7] J. Veladimirov, “Fundamental Physics,” Philosophy and Religion, Kostroma, 1996.

[8] E. Naschie, “On a Class of General Theories for High Energy Particle Physics,” Chaos, Solitons and Fractals, Vol. 14, No. 4, 2005, pp. 649-680. doi:10.1016/S0960-0779(02)00033-4

[9] V. V. Petrunenko, “The Golden Section of Quantom States and Its Astronomical and Phisical Manifestations,” Pravo i economika, 2005.

[10] Y. S. Vladimirov, “Metaphysics,” Binom, Moscow, 2002.

[11] E. Kilic and A. P. Stakhov, “On the Fibonacci and Lucas p-numbers, Their Sums, Families of Bipartite Graphs and Permanents of Certain Matrices,” Chaos, Solitons and Fractals, Vol. 40, No. 5, 2009, pp. 2210-2221. doi:10.1016/j.chaos.2007.10.007

[12] A. Stakhov and B. Rozin, “The Continuous Functions for the Fibonacci and Lucas P-Numbers,” Chaos, Solitons and Fractals, Vol. 28, No. 4 , 2006, pp. 1014-1025. doi:10.1016/j.chaos.2005.08.158

[13] M. Akbulak and D. Bozkurt, “On the Order-M Generalized Fibonacci K-Numbers,” Chaos, Solitons and Fractals, Vol. 42, No. 3, 2009, pp. 1347-1355. doi:10.1016/j.chaos.2009.03.019

[14] A. Stakhov and B. Rozin, “Theory of Binet for Fibonacci and Lucas p-Numbers,” Chaos, Solitons and Fractals, Vol. 27, No. 5, 2006, pp. 1162-1177. doi:10.1016/j.chaos.2005.04.106

[15] A. Nalli and P. Haukkanen, “On Generalized Fibonacci and Lucas Polynomials,” Chaos, Solitons and Fractals, Vol. 42, No. 5, 2009, pp. 3179-3186. doi:10.1016/j.chaos.2009.04.048

[16] F. Buyukkilic and D. Demirhan, “Cumulative Growth with Fibonacci Approach, Golden Section and Physics,” Chaos, Solitons and Fractals, Vol. 42, No. 1, 2009, pp. 24-32. doi:10.1016/j.chaos.2008.10.023

[17] A. Benavoli, L. Chisci and A. Farina, “Fibonacci Sequence, Golden Section, Kalman Filter and Optimal Control,” Signal Processing, Vol. 89, No. 8, 2009, pp. 1483-1488. doi:10.1016/j.sigpro.2009.02.003

[18] E. Cureg and A. Mukherjea, “Numerical Results on Some Generalized Random Fibonacci Sequences,” Computer and Mathematics Application, Vol. 59, No. 1, 2010, pp. 233-246. doi:10.1016/j.camwa.2009.08.001

[19] E. M. Soroko, “Structural Harmony of System,” Nauka i Tekhnika, Russian, 1984.

[20] A. Stakhov and B. Rozin, “The Golden Shofar,” Chaos, Solitons and Fractals, Vol. 26, No. 3, 2005, pp. 677-684. doi:10.1016/j.chaos.2005.01.057

[21] A. P. Stakhov, “Generalized Golden Section and a New Approach to Geometric Definition of a Number,” Ukrainian Mathematical Journal, Vol. 56, No. 8, 2004, pp. 1143-1150. doi:10.1007/s11253-005-0064-3

[22] O. Karpenkov, “On Examples of Two-Dimensinal Priodic Continued Fractions,” Academic Paper, Moscow State University, Moscow, 2004.

[23] M. Grundland, J. Patera, Z. Masakova and N. Dodgson, “Image Sampling with Quasicrystals,” Electrical and Electronic Engineering, Vol. 5, 2009, pp. 1-23. doi: 10.3842/SIGMA.2009.075

[24] S. Dutch, “Why Trisecting of the Angle Is Impossible?” Natural and Applied Sciense, University of Wisconsin-Green Bay, Green Bay, 2009.

[25] J. B. Fraleigh, “A First Course in Abstract Algebra,” 7th Edition, Addison-Wesley, Boston, 1982.

[26] M. B. Farnworth, “Cubic Equations and Ideal Trisection of the Arbitrary Angle,” Theaching Mathematics and Its Applications, Vol. 25, No. 2, 2006.

[27] S. Falcon and A. Plaza, “On the Fibonacci K-Numbers,” Chaos, Solitons and Fractals, Vol. 32, No. 5, 2007, pp. 1615-1624. doi:10.1016/j.chaos.2006.09.022

[28] A. Stakhov, “The Golden Section, Secret of the Egyption Civilization and Harmony Mathematics,” Chaos, Solitons and Fractals, Vol. 30, No. 2, 2006, pp. 490-505. doi:10.1016/j.chaos.2005.11.022

[1] D. H. Fowler, “A Generalization of the Golden Section,” Fibonacci Quarterly, Vol. 20, 1982, pp. 146-158.

[2] A. P. Stakhov, “The Generalized Principle of the Golden Section and Its Applications in Mathematics, Science and Engineering,” Chaos, Solitons and Fractals, Vol. 26, No. 2, 2005, pp. 263-289. doi:10.1016/j.chaos.2005.01.038

[3] A. Stakhov, “Fundamentals of a New Kind of Mathematics Based on the Golden Section,” Chaos, Solitons and Fractals, Vol. 27, No. 5, 2006, pp. 1124-1146. doi:10.1016/j.chaos.2005.05.008

[4] M. Basu and B. Prasad, “Coding Theory on the M-Extension of the Fibonacci P-Numbers,” Chaos, Solitons and Fractals, Vol. 42, No. 4, 2009, pp. 2522-2530. doi:10.1016/j.chaos.2009.03.197

[5] M. Basu and B. Prasad, “The Generalized Relations among the Code Elements for Fibonacci Coding Theory,” Chaos, Solitons and Fractals, Vol. 41, No. 5, 2009, pp. 2517-2525. doi:10.1016/j.chaos.2008.09.030

[6] E. G. Kocer, N. Tuglu and A. Stakhov, “On the M-Extension of the Fibonacci and Lucas P-Numbers, Chaos,” Chaos, Solitons and Fractals, Vol. 40, No. 4, 2009, pp. 1890-1906. doi:10.1016/j.chaos.2007.09.071

[7] J. Veladimirov, “Fundamental Physics,” Philosophy and Religion, Kostroma, 1996.

[8] E. Naschie, “On a Class of General Theories for High Energy Particle Physics,” Chaos, Solitons and Fractals, Vol. 14, No. 4, 2005, pp. 649-680. doi:10.1016/S0960-0779(02)00033-4

[9] V. V. Petrunenko, “The Golden Section of Quantom States and Its Astronomical and Phisical Manifestations,” Pravo i economika, 2005.

[10] Y. S. Vladimirov, “Metaphysics,” Binom, Moscow, 2002.

[11] E. Kilic and A. P. Stakhov, “On the Fibonacci and Lucas p-numbers, Their Sums, Families of Bipartite Graphs and Permanents of Certain Matrices,” Chaos, Solitons and Fractals, Vol. 40, No. 5, 2009, pp. 2210-2221. doi:10.1016/j.chaos.2007.10.007

[12] A. Stakhov and B. Rozin, “The Continuous Functions for the Fibonacci and Lucas P-Numbers,” Chaos, Solitons and Fractals, Vol. 28, No. 4 , 2006, pp. 1014-1025. doi:10.1016/j.chaos.2005.08.158

[13] M. Akbulak and D. Bozkurt, “On the Order-M Generalized Fibonacci K-Numbers,” Chaos, Solitons and Fractals, Vol. 42, No. 3, 2009, pp. 1347-1355. doi:10.1016/j.chaos.2009.03.019

[14] A. Stakhov and B. Rozin, “Theory of Binet for Fibonacci and Lucas p-Numbers,” Chaos, Solitons and Fractals, Vol. 27, No. 5, 2006, pp. 1162-1177. doi:10.1016/j.chaos.2005.04.106

[15] A. Nalli and P. Haukkanen, “On Generalized Fibonacci and Lucas Polynomials,” Chaos, Solitons and Fractals, Vol. 42, No. 5, 2009, pp. 3179-3186. doi:10.1016/j.chaos.2009.04.048

[16] F. Buyukkilic and D. Demirhan, “Cumulative Growth with Fibonacci Approach, Golden Section and Physics,” Chaos, Solitons and Fractals, Vol. 42, No. 1, 2009, pp. 24-32. doi:10.1016/j.chaos.2008.10.023

[17] A. Benavoli, L. Chisci and A. Farina, “Fibonacci Sequence, Golden Section, Kalman Filter and Optimal Control,” Signal Processing, Vol. 89, No. 8, 2009, pp. 1483-1488. doi:10.1016/j.sigpro.2009.02.003

[18] E. Cureg and A. Mukherjea, “Numerical Results on Some Generalized Random Fibonacci Sequences,” Computer and Mathematics Application, Vol. 59, No. 1, 2010, pp. 233-246. doi:10.1016/j.camwa.2009.08.001

[19] E. M. Soroko, “Structural Harmony of System,” Nauka i Tekhnika, Russian, 1984.

[20] A. Stakhov and B. Rozin, “The Golden Shofar,” Chaos, Solitons and Fractals, Vol. 26, No. 3, 2005, pp. 677-684. doi:10.1016/j.chaos.2005.01.057

[21] A. P. Stakhov, “Generalized Golden Section and a New Approach to Geometric Definition of a Number,” Ukrainian Mathematical Journal, Vol. 56, No. 8, 2004, pp. 1143-1150. doi:10.1007/s11253-005-0064-3

[22] O. Karpenkov, “On Examples of Two-Dimensinal Priodic Continued Fractions,” Academic Paper, Moscow State University, Moscow, 2004.

[23] M. Grundland, J. Patera, Z. Masakova and N. Dodgson, “Image Sampling with Quasicrystals,” Electrical and Electronic Engineering, Vol. 5, 2009, pp. 1-23. doi: 10.3842/SIGMA.2009.075

[24] S. Dutch, “Why Trisecting of the Angle Is Impossible?” Natural and Applied Sciense, University of Wisconsin-Green Bay, Green Bay, 2009.

[25] J. B. Fraleigh, “A First Course in Abstract Algebra,” 7th Edition, Addison-Wesley, Boston, 1982.

[26] M. B. Farnworth, “Cubic Equations and Ideal Trisection of the Arbitrary Angle,” Theaching Mathematics and Its Applications, Vol. 25, No. 2, 2006.

[27] S. Falcon and A. Plaza, “On the Fibonacci K-Numbers,” Chaos, Solitons and Fractals, Vol. 32, No. 5, 2007, pp. 1615-1624. doi:10.1016/j.chaos.2006.09.022

[28] A. Stakhov, “The Golden Section, Secret of the Egyption Civilization and Harmony Mathematics,” Chaos, Solitons and Fractals, Vol. 30, No. 2, 2006, pp. 490-505. doi:10.1016/j.chaos.2005.11.022