Hyperbolic Fibonacci and Lucas Functions

Let's remember the definition of the hyperbolic functions , the hyperbolic sine

(1)

and the hyperbolic cosine

(2)

And let's remember now the definition of Binet's formulas - for Fibonacci numbers

(3)
(4)

and for Lucas numbers

(5)
(6)

where k = 0, ±1, ±2, ±3, ... .

Formulas (3)-(6) were proved by the French 19th century mathematician Binet but they are so much excellent since mathematical point of view that until now they call a feeling of great admiration and deep respect. In fact, it is impossible to suppose that integral numbers (Fibonacci and Lucas numbers) can be presented as a difference or a sum of two irrational numbers (powers of the golden ratio) - for Lucas numbers and a sum or a difference of two irrational numbers divided by the irrational numbers . But Binet's formulas (3)-(6) show that this is possible for all Fibonacci and Lucas series.

And now let's compare formulas (3)-(6) with hyperbolic functions (1) and (2). We can see that formulas (3) and (5) are similar with the hyperbolic sine (1) but formulas (4) and (6) are similar with the hyperbolic cosine (2). A certain external similarity of Binet's formulas (3) - (6) with the classical hyperbolic functions (1), (2) is a basis for introduction of the Fibonacci and Lucas hyperbolic functions. With this in mind let us replace the discrete variable k in formulas (3) - (6) by the continues variable x and introduce the following continues functions:

(1) Fibonacci hyperbolic sine
(7)

(2) Fibonacci hyperbolic cosine
(8)

(3) Lucas hyperbolic sine
(9)

(4) Lucas hyperbolic cosine
(10)

Note that for the discrete values x = k the Fibonacci and Lucas hyperbolic functions are coincident with the Fibonacci and Lucas numbers, i. e.

sFk = F2k;   cFx = F2k+1;   sLx = L2k+1;   cLx = L2k.(11)

Probably the Ukrainian scientists A.P. Stakhov and I.S. Tkachenko were the former scientists who had came to the idea of the Fibonacci and Lucas hyperbolic functions. The theory of new classes of hyperbolic functions is stated in their article "Hyperbolic Fibonacci trigonometry" published in the "Reports of the Ukrainian Academy of Sciences" (1993, V. 208, No 7).

Thus the main result following from our very simple consideration is introducing a new classes of elementary functions, a class of Fibonacci hyperbolic functions and a class of Lucas hyperbolic functions. These functions are too similar to the classical hyperbolic functions but differ from them by one peculiarity. In contrast to the classical hyperbolic functions the new hyperbolic functions have a numerical analogy, Fibonacci numbers - for Fibonacci hyperbolic functions and Lucas numbers - for Lucas hyperbolic functions.

Of what importance have new classes of the hyperbolic functions for general science and mathematics, in particular?

Let us begin from the theory of Fibonacci numbers. Until now the theory of Fibonacci numbers develops as discrete theory because Fibonacci numbers are a part of natural numbers and belong to the discrete set. But the Fibonacci and Lucas hyperbolic functions are "continues" mathematical objects and we can apply methods of "continues" mathematics to investigate these functions. But each mathematical identity for the Fibonacci and Lucas hyperbolic functions has a "Fibonacci" interpretation using (11). Such approach allows to develop the Fibonacci numbers theory as "continues" theory.

But a new theory of phyllotaxis developed by the Ukrainian architect Oleg Bodnar is the best evidence of the Fibonacci and Lucas functions effectiveness for simulation of natural phenomena. In his book "Golden Section and Non-Euclidean Geometry in Nature and Art" (1994)

Oleg Bodnar's book 'Golden Section and Non-Euclidean Geometry in Nature and Art' (1994)

Using the Fibonacci hyperbolic functions Bodnar proved that Fibonacci numbers arising at the surface of the phyllotaxis objects are a consequence of hyperbolic character of growth processes of the objects. But the living nature uses the Fibonacci hyperbolic functions for construction of its objects and this fundamental fact is confirmed with the "phyllotaxis laws" based on Fibonacci numbers!

Follow us! We will tell about Bodnar's discovery at the corresponding page of our Museum.