Making use of this qanalogue of the golden ratio, we defined sin fqhx and cos fqhx functions. Generalized fibonacci sequences and its properties 143 in this paper, we present properties of generalized fibonacci sequences like catalans identity, cassinis identity and docagness identity. Nevanlinna extremal measures for polynomials related to qfibonacci polynomials frantisek stampach faculty of information technology, ctu in prague symmetries of discrete systems and processes. The 3dimensional kfibonacci spirals are studied from a geometric point of view. In table 2 some formulas for the classical hyperbolic functions and the corresponding formulas for the hyperbolic fibonacci functions are represented. From q2 n qnq nd a formula for the sum of squares of two consecutive fibonacci numbers.
In this article i have included some graphic formats such as fibonacci arcs, fan, channel, expansion, wich are created also with fibonacci retracement and also rules to perfect chart plotting. Pdf on the properties of kfibonacci numbers semantic scholar. W e shall introduce new functions called fibonacci hyperbolic functions and show how they result from suitable application of m odified num erical triangles. It provides a broad introduction to the fascinating and beautiful subject of the mathematics of harmony, a new interdisciplinary direction of modern science.
On the properties of kfibonacci numbers 1 introduction. Hyperbolic jacobsthal numbers asian research journal of. The corresponding code matrix will prevent the attack to the. Bifurcations of fibonacci generating functions sciencedirect. Fibonacci hyperbolic functions from wolfram mathworld. It has been emphasized by 7 that the fibonacci oscillators o er a uni cation of quantum oscillators related to quantum groups 2. Wilkertype inequalities for hyperbolic fibonacci functions. In this paper, we obtain some new identities for kfibonacci numbers. He wrote several mathematical texts that, among other things, introduced europe. The hyperbolic fibonacci function, which is the being extension of binets formula for the fibonacci number in continuous domain, transform the fibonacci number theory into continuous theory. How to use fibonacci retracement to predict forex market. The using the results of theorem 9, we obtain huygens, wilker, sandorbencze, carlson, and shaferfinktype inequalities respectively. In this paper we introduce the hyperbolic kpell functions and new classes of quaternions associated with this type of functions are presented. In addition, the binet formulas, generating functions and some properties of these functions and quaternions sequences are studied.
Hyperbolic expressions of polynomial sequences and parametric. Quaternions, hyperbolic functions, kpell sequence, binets. To show the influence of small q deviations on the bifurcation diagram we have numerically calculated the bifurcation diagram for the following values of q 1. I covered hyperbolic trigonometric functions in a recent maths course. Some properties of the p, q fibonacci and p, q lucas polynomials lee, gwangyeon and asci, mustafa, journal of applied mathematics, 2012. The construction of numerical algorithms is described and commented. In order to prove the claim, we consider the equation 4.
Hyperbolic fibonacci and lucas functions scientific research. The fibonacci hyperbolic functions sfh x left and cfh x right. The 3dimensional k fibonacci spirals are studied from a geometric point of view. In the material below i have tried to explain how can be used fibonacci retracement as an important tool to predict forex market. Nevanlinna extremal measures for polynomials related to q.
As a special case, some famous identities like cassinis, askeys are derived and presented, and also a new class of multi parameters hyperbolic functions and their properties are introduced, finally a generalized qmatrix called gnmatrix of order n being a generating matrix for the generalized fibonacci numbers of order n and its inverse. Using the definitions of qanalogue of pell hyperbolic sine and cosine functions. Moreover, we investgate lorentzian inner product for the hyperbolic jacobsthal vectors. We introduce the qanalogue of classical golden ratio as follow. Request pdf the kfibonacci hyperbolic functions an extension of the. Hyperbolic fibonacci and lucas functions, golden fibonacci. Shiue, yand tsuiwei weng z august 5, 2012 abstract a sequence of polynomial fa nxgis called a function sequence of order 2 if it satis es the linear recurrence relation of order 2. However i was never presented with any reasons as to why or even if they are useful. I j good complex fibonacci and lucas numbers, continued fractions, and the square root of the golden ratio, fib q 31 1993 pages 719. It is proved 2, 3 that for each identity for classical hyperbolic functions there is an analog in the form of the corresponding identity for the hyperbolic fibonacci and lucas functions. The nth fibonacci number fn is the nearest whole number to. The qfibonacci hyperbolic functions semantic scholar.
He traveled extensively in europe and northern africa. Now, equalities introduced 12 can naturally be related with the qfibonacci hyperbolic functions studied before. It turns out that similar standard matrix properties lead to corresponding fibonacci results. In particular, we shall analyze functions classified by criterion k that is the number following from the method of geometric interpretation of hyperbolic. These curves appear naturally from studying the k fibonacci numbers ff k. Multi parameters golden ratio and some applications. In this paper, several properties of these kfibonacci hyperbolic functions are studied in an easy way. The q series thus obtained are then expressed in closed form in terms of complete elliptic integrals of the first and second kind and the corresponding modulus. Complex factorizations of the lucas sequences via matrix methods wu, honglin, journal of applied mathematics, 2014. W e shall also establish a set of suitable properties of f ibonacci hyperbolic functions such as sym m etry, shifting, and links w ith the.
This drives the price lower until such a point that the bulls, sensing the price is better value, enter. The following correlations that are similar to the equation coshx2. The main result of reference 12 is a proof of the following expression for the. Based on an analogy between the binet formula, and the classical hyperbolic functions. Pdf the kfibonacci hyperbolic functions angel plaza. On chebyshev polynomials, fibonacci polynomials, and. Symmetric fibonacci function solutions of some nonlinear. Some remarkable infinite product identities involving fibonacci and. We shall introduce new functions called fibonacci hyperbolic functions and show. The qpell hyperbolic functions natural sciences publishing. Recently the new unique classes of hyperbolic functionshyperbolic fibonacci functions based on the golden ratio, and hyperbolic fibonacci lfunctions based on the metallic proportions l is a given natural number, were introduced in mathematics. This volume is a result of the authors four decades of research in the field of fibonacci numbers and the golden section and their applications. We have to remark that even small deviations of q from the value q 1 are changing this picture dramatically.
We only give the proof of the 14, because the proof of. Technical analysis fibonacci levels retracements a retracement is a pullback within the context of a trend. By comparing binet formulas for the fibonacci and lucas numbers to the formulas of classical hyperbolic matrix functions, we will define hyperbolic fibonacci matrix functions and we will deal with. Hyperbolic fibonacci and lucas functions and a new comprehension of the golden mean role in modern science unfortunately, mathematicians of 19th and 20th century could not evaluate the true value of binet formulas, although these formulas contained a hint on the important mathematical discoveryhyperbolic fibonacci and lucas functions. The graph of coshx is always above the graphs of ex2 and e. The introduced above fibonacci hyperbolic functions is connected with the hyperbolic functions by the following simple correlations 12. The graphics of these new kfibonacci hyperbolic functions are shown in fig. The generalized fibonacci hyperbolic functions cfhqk and sfh q k can be regarded as generating functions for polynomials t k q and p k q, k 0, 1, 2. The golden section, fibonacci series, and new hyperbolic models. The graphics of these new k fibonacci hyperbolic functions are shown in fig. These functions can be regarded as q extensions of classical hyperbolic functions. We finalize with the introduction of some curves and. Multi parameters golden ratio and some applications applied.
Stakhov and tkachenko introduced a new class of hyperbolic functions called hyperbolic fibonacci functions replacing the discrete variable n in equation with the continuous variable x that takes its values from the set of real numbers. In this paper, we introduce the hyperbolic jacobsthal numbers and we present recurrence relations, binets formulas, generating functions and the summation formulas for these numbers. Leonardo pisano fibonacci was born around 1170 and died around 1250 in pisa in what is now italy. A complete characterization of bpower fractions that can be represented as series of general nbonacci numbers jz lee, js lee fibonacci quarterly 25 1987 pages 7275. These curves appear naturally from studying the kfibonacci numbers ff k. Chapter 2 fibonacci numbers fibonacci numbers introduce vectors, functions and recursion. Real world uses of hyperbolic trigonometric functions. Pdf the golden section, fibonacci series, and new hyperbolic. In this paper, we study qanalogue of fibonacci hyperbolic functions. The aim of this study to give q analogue of the pell hyperbolic functions. Dip after a rise from 0 to 1, short term market participants start to take profit.
The qseries thus obtained are then expressed in closed form in terms of complete elliptic integrals of the first and second kind and the corresponding modulus. A method of summing sixteen series of hyperbolic functions is presented. The kfibonacci hyperbolic functions request pdf researchgate. Siyavulas open mathematics grade 11 textbook, chapter 5 on functions covering hyperbolic functions. Since r1 and r2 are the roots of the characteristic equation eq. Number theory special numbers fibonacci numbers calculus and analysis special functions miscellaneous special functions fibonacci hyperbolic functions. The summation of series of hyperbolic functions siam.
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