2.7. Fibonacci numbers

In the 1202 AD, Leonardo Fibonacci wrote in his book Liber Abaci of a simple numerical sequence that is the foundation for an incredible mathematical relationship behind phi. This sequence was known as early as the sixth century AD by Indian mathematicians, but it was Fibonacci who introduced it to the west after his travels throughout the Mediterranean world and North Africa. He is also known as Leonardo Bonacci, as his name is derived in Italian from words meaning son of (the) Bonacci.

So, in the first month, we have only the first pair of rabbits. Likewise, in the second month, we again have only our initial pair of rabbits. However, by the third month, the pair will give birth to another pair of rabbits, and there will now be two pairs. Continuing on, we find that in month 4, we will have 3 pairs, then 5 pairs in month 5, then 8, 13, 21, 34, …, etc., continuing in this manner. It is quite apparent that this sequence directly corresponds with the Fibonacci sequence introduced above, and indeed, this is the first problem ever associated with the now-

Nature of Phyllotaxy and Topology of H-matrix http://dx.doi.org/10.5772/intechopen.74676 79

Fibonacci numbers have many interesting properties and applications in arts, sciences and

<sup>f</sup> <sup>k</sup> <sup>¼</sup> <sup>f</sup> <sup>n</sup>þ<sup>2</sup> � <sup>1</sup>; n<sup>∈</sup> <sup>N</sup>,

<sup>k</sup> <sup>¼</sup> <sup>f</sup> <sup>n</sup> <sup>f</sup> <sup>n</sup>þ<sup>1</sup>; n<sup>∈</sup> <sup>N</sup>

Everything in Nature is subordinated to stringent mathematical laws. Prove to be that leaf's disposition on plant's stems also has stringent mathematical regularity and this phenomenon is called phyllotaxis in botany. An essence of phyllotaxis consists in a spiral disposition of leaves on plant's stems of trees, petals in flower baskets, seeds in pine cone and sunflower

This phenomenon, known already to Kepler, was a subject of discussion of many scientists, including Leonardo da Vinci, Turing, Veil, and so on. In phyllotaxis phenomenon, more complex concepts of symmetry, in particular, a concept of helical symmetry, are used. The phyllotaxis phenomenon reveals itself especially brightly in inflorescences and densely packed botanical structures such as pine cones, pineapples, cacti, heads of sunflower and cauliflower,

On the surfaces of such objects, their bio-organs (seeds on the disks of sunflower heads and pine cones, etc.) are placed in the form of the left-twisted and right-twisted spirals. For such

architecture. Also, following [7], some basic properties are as follows

Xn k¼0

> Xn k¼0 f 2

famous numbers.

and

head, etc.

and many other objects [11].

The Fibonacci numbers have been introduced [7–14]. The Fibonacci numbers are the sequence of numbers f <sup>n</sup> , n∈ N defined by recurrence relations

$$f\_0 = 0, f\_1 = 1 \text{ and } f\_n = f\_{n-1} + f\_{n-2}; n \ge 2$$

First derived from the famous rabbit problem of 1228, the Fibonacci numbers were originally used to represent the number of pairs of rabbits born of one pair in a certain population. Let us assume that a pair of rabbits is introduced into a certain place in the first month of the year. This pair of rabbits will produce one pair of offspring every month, and every pair of rabbits will begin to reproduce exactly 2 months after being born. No rabbit ever dies, and every pair of rabbits will reproduce perfectly on schedule.


So, in the first month, we have only the first pair of rabbits. Likewise, in the second month, we again have only our initial pair of rabbits. However, by the third month, the pair will give birth to another pair of rabbits, and there will now be two pairs. Continuing on, we find that in month 4, we will have 3 pairs, then 5 pairs in month 5, then 8, 13, 21, 34, …, etc., continuing in this manner. It is quite apparent that this sequence directly corresponds with the Fibonacci sequence introduced above, and indeed, this is the first problem ever associated with the nowfamous numbers.

Fibonacci numbers have many interesting properties and applications in arts, sciences and architecture. Also, following [7], some basic properties are as follows

$$\sum\_{k=0}^{n} f\_k = f\_{n+2} - 1; n \in \mathbb{N}\_{\ge 0}$$

and

2.7. Fibonacci numbers

78 Matrix Theory-Applications and Theorems

of numbers f <sup>n</sup>

words meaning son of (the) Bonacci.

of rabbits will reproduce perfectly on schedule.

In the 1202 AD, Leonardo Fibonacci wrote in his book Liber Abaci of a simple numerical sequence that is the foundation for an incredible mathematical relationship behind phi. This sequence was known as early as the sixth century AD by Indian mathematicians, but it was Fibonacci who introduced it to the west after his travels throughout the Mediterranean world and North Africa. He is also known as Leonardo Bonacci, as his name is derived in Italian from

The Fibonacci numbers have been introduced [7–14]. The Fibonacci numbers are the sequence

<sup>f</sup> <sup>0</sup> <sup>¼</sup> <sup>0</sup>, f <sup>1</sup> <sup>¼</sup> 1 and <sup>f</sup> <sup>n</sup> <sup>¼</sup> <sup>f</sup> <sup>n</sup>�<sup>1</sup> <sup>þ</sup> <sup>f</sup> <sup>n</sup>�<sup>2</sup>; n <sup>≥</sup> <sup>2</sup>

First derived from the famous rabbit problem of 1228, the Fibonacci numbers were originally used to represent the number of pairs of rabbits born of one pair in a certain population. Let us assume that a pair of rabbits is introduced into a certain place in the first month of the year. This pair of rabbits will produce one pair of offspring every month, and every pair of rabbits will begin to reproduce exactly 2 months after being born. No rabbit ever dies, and every pair

, n∈ N defined by recurrence relations

$$\sum\_{k=0}^{n} f\_k^2 = f\_n f\_{n+1}; n \in \mathbb{N}$$

Everything in Nature is subordinated to stringent mathematical laws. Prove to be that leaf's disposition on plant's stems also has stringent mathematical regularity and this phenomenon is called phyllotaxis in botany. An essence of phyllotaxis consists in a spiral disposition of leaves on plant's stems of trees, petals in flower baskets, seeds in pine cone and sunflower head, etc.

This phenomenon, known already to Kepler, was a subject of discussion of many scientists, including Leonardo da Vinci, Turing, Veil, and so on. In phyllotaxis phenomenon, more complex concepts of symmetry, in particular, a concept of helical symmetry, are used. The phyllotaxis phenomenon reveals itself especially brightly in inflorescences and densely packed botanical structures such as pine cones, pineapples, cacti, heads of sunflower and cauliflower, and many other objects [11].

On the surfaces of such objects, their bio-organs (seeds on the disks of sunflower heads and pine cones, etc.) are placed in the form of the left-twisted and right-twisted spirals. For such

phyllotaxis objects, it is used usually the number ratios of the left-hand and right-hand spirals observed on the surface of the phyllotaxis objects. Botanists proved that these ratios are equal to the ratios of the adjacent Fibonacci numbers, that is,

sequence x ¼ ð Þ xk ∈ X the sequence Ax ¼ ð Þ Ax <sup>n</sup>

to l if Ax converges to l which is called as the A-limit of x.

ð Þ Ax <sup>n</sup> <sup>¼</sup> <sup>X</sup>

For a sequence space X, the matrix domain XA of an infinite matrix A is defined as

k

For simplicity in notation, here and in what follows, the summation without limits runs from 0 to ∞. By ð Þ X; Y , we denote the class of all such matrices. A sequence x is said to be A-summable

An infinite matrix A ¼ ð Þ ank is said to be regular if and only if the following conditions (or

ukf <sup>k</sup> 2

8 ><

>:

<sup>f</sup> <sup>n</sup> <sup>f</sup> <sup>n</sup>þ<sup>1</sup>

0, if k > n:

10000 ⋯ 1=2 1=20 0 0 ⋯ 1=6 1=6 4=60 0 ⋯ 1=15 1=15 4=15 9=15 0 ⋯ ⋮ ⋮ ⋮ ⋮ ⋱⋮

n∈ N. Also, since it satisfies the conditions of Toeplitz matrix and hence it is regular matrix.

Qn <sup>¼</sup> <sup>X</sup><sup>n</sup> k¼0 f 2

<sup>k</sup> , then the matrix <sup>H</sup> is special case of the matrix Rq

<sup>k</sup> <sup>¼</sup> <sup>f</sup> <sup>n</sup> <sup>f</sup> <sup>n</sup>þ<sup>1</sup>,

if 0 ≤ k ≤ n,

ankxk:

where

i. lim<sup>n</sup>!<sup>∞</sup>

iii. <sup>P</sup><sup>∞</sup> k¼0

which is a sequence space.

X∞ k¼0

Toplitz conditions) hold [17–19]:

ank ¼ 1,

ii. lim<sup>n</sup>!<sup>∞</sup> ank <sup>¼</sup> <sup>0</sup>, kð Þ <sup>¼</sup> <sup>0</sup>; <sup>1</sup>; <sup>2</sup>;… ,

∣ank∣ < M, Mð Þ > 0; j ¼ 0; 1; 2;… :

Thus, for uk ¼ 1 and for all k∈ N, we have

Note that if we take qk ¼ f

introduced by Sheikh and Ganie [16].

In the present paper, we introduce <sup>H</sup>-matrix with <sup>H</sup> <sup>¼</sup> <sup>h</sup><sup>u</sup>

H ¼

It is obvious that the matrix H is a triangle, that is, hu

2

0

BBBBBB@

hu nk ¼ � �, the A-transform of x exists and is in Y

Nature of Phyllotaxy and Topology of H-matrix http://dx.doi.org/10.5772/intechopen.74676 81

XA ¼ f g x ¼ ð Þ xk ∈ ω : Ax∈ X , (1)

nk � � n, k<sup>∈</sup> <sup>N</sup> as follows:

1

CCCCCCA :

nk ¼ 0 for k > n and for all

<sup>u</sup>, where

nn 6¼ 0 and hu

$$\frac{f\_{i+1}}{f\_i} : \frac{2}{1}, \frac{3}{2}, \frac{5}{3}, \frac{8}{5}, \frac{13}{8}, \dots = \frac{1+\sqrt{5}}{2}$$

By using hyperbolic Fibonacci functions, he had developed an original geometric theory of phyllotaxis and explained why Fibonacci spirals arise on the surface of the phyllotaxis objects namely, pine cones, cacti, pine apple, heads of sunflower, and so on, in process of their growths. Bodnar's geometry [15] confirms that these functions are 'natural' functions of the nature, which show their value in the botanic phenomenon of phyllotaxis. This fact allows us to assert that these functions can be attributed to the class of fundamental mathematical discoveries of contemporary science because they reflect natural phenomena, in particular, phyllotaxis phenomenon.

From above discussion, it gave us motivation to see the behavior of the infinite matrices generated by Fibonacci numbers.

In the present chapter, we have introduced a new type of matrix <sup>H</sup> <sup>¼</sup> hu nk � � n, k<sup>∈</sup> <sup>N</sup> by using Fibonacci numbers f <sup>n</sup> and we call it as H-matrix generated by Fibonacci numbers f <sup>n</sup> and introduce some new sequence spaces related to matrix domain of H in the sequence spaces lp, l∞, c and c0, where 1 ≤ p < ∞.
