Polynomials Related to Generalized Fibonacci Sequence

Abstract

The Fibonacci polynomials are a polynomial sequence that can be considered as a generalization of the Fibonacci numbers. Fibonacci polynomials are defined by a recurrence relation: Fnx=xFn−1x+Fn−2x,n≥2 where F0=0,F1=1. The first few Fibonacci polynomials are F0=0, F0x=0, F1x=1, F2x=x, F3x=x2+1. In this chapter, we extend the Fibonacci recurrence relation to define the sequence {Kn} and will derive some properties of this sequence. We also define four comparison sequences {Pn}, {Qn}, {Rn}, and {Sn} and obtain some identities with the help of generating matrix.

1. Introduction

The Fibonacci sequence [1] receives its name from Leonardo Pisano, known as Fibonacci who was the most talented Italian mathematician of middle age. It is supposed that he was the first mathematician who introduced the Hindu-Arabic system of numbers to Italians. His work ‘Liber-Abaci’ (1202) is famous for this.

In the Liber Abaci, Leonardo states the famous “Rabbit Problem” for attaining the output of this rabbit problem.

1.1 Utilization of Fibonacci sequence in the study of famous rabbit problem

“How many pairs of rabbits are born of one pair in a year?” This problem is stated in the form: “Suppose a newly-born pair of rabbits, one male and one female, are put in a field. Rabbits are able to mate at the age of 1 month so that at the end of its second month a female can produce another pair of rabbits.”

Suppose that our rabbits never die and that the female always produces one new pair (one male and one female) every month from the second month on.

Leonardo also gave the solution to this problem and obtained the sequence of numbers as a result:

$$1,1,2,3,5,8,\dots$$

This sequence is called the Fibonacci sequence. The Fibonacci sequence is defined by the recurrence relation as,

$$F_n=F_{n-1}+F_{n-2},n>1$$

Waddilli, M.E. [2] has extended the Fibonacci recurrence relation to define the sequence {$K_n$}, where,

$$K_n=K_{n-1}+K_{n-2}+K_{n-3},n>3$$E1

where, $K_0,K_1,K_2$ are given arbitrary algebraic integers.

Jaiswal, D.V. [3] has extended Fibonacci recurrence relation to define the sequence {$Q_0$}, where,

$$Q_n=Q_{n-1}+Q_{n-2}+Q_{n-3}+Q_{n-4},n>4$$E2

where, $Q_0,Q_1,Q_2$ are given arbitrary algebraic integers.

Harne, S. [4] has extended Fibonacci recurrence relation to define the sequence {$D_n$}, where,

$$D_n=D_{n-1}+D_{n-2}+D_{n-3}+D_{n-4}+Q_{n-5},n>5$$E3

where, $D_0,D_1,D_2$ are given arbitrary algebraic integers.

In this chapter, Teeth MS. [5] shall further extend the Fibonacci recurrence relation [6, 7, 8, 9, 10] to define the sequence {$C_n$} and shall discuss some properties of this sequence. We shall also consider the four comparison sequences {$P_n$}, {$Q_n$}, {$R_n$}, and {$K_n$}.

2. The generalized sequence as per our propose model {$K_n$}

We consider the following sequence,

$$\left\{C_n\right\}=C_0,C_1,C_2,C_3,\dots .,C_n$$

where, $C_0,C_1,C_2,C_3,C_4,C_5,$C₀ are arbitrary algebraic integers all of which are not zero and

$$C_n=C_{n-1}+C_{n-2}+C_{n-3}+C_{n-4}+C_{n-5}+C_{n-6},n\ge 6$$E4

We also consider the sequence $\left\{P_n\right\}=P_0,P_1,P_2,P_3,\dots .,P_{n.}$

where,

$$\begin{array}{c}{P}_0=C_3-C_2-C_1-C_0\\ P_1=C_4-C_3-C_2-C_1\\ P_2=C_5-C_4-C_3-C_2\\ P_3=C_6-C_5-C_4-C_3\\ P_4=C_7-C_6-C_5-C_4\end{array}$$E5

$$\text{with},P_n=C_{n-1}+C_{n-2}+C_{n-3}+C_{n-4}+C_{n-5},n\ge 5$$E6

and $\left\{Q_n\right\}=Q_0,Q_1,Q_2,Q_3,\dots \dots .,Q_n$

where,

$$\begin{array}{c}{Q}_0=C_4-C_3-C_2-C_1\\ Q_1=C_5-C_4-C_3-C_2-C_1\\ Q_2=C_6-C_5-C_4-C_3-C_2\end{array}$$E7

$$\text{with},Q_n=C_{n-1}+C_{n-2}+C_{n-3}+C_{n-4}$$E8

and $\left\{R_n\right\}=R_0,R_1,R_2,R_3,\dots .,R_n$where,

$$\begin{array}{c}{R}_0=C_5-C_4-C_3-C_2-C_1-C_0\\ R_1=C_6-C_5-C_4-C_3-C_2-C_1\\ R_2=C_7-C_6-C_5-C_4-C_3-C_2\\ R_3=C_8-C_7-C_6-C_5-C_4-C_3\\ R_4=C_9-C_8-C_7-C_6-C_5-C_4\end{array}$$E9

$$\text{with},R_n=C_{n-1}+C_{n-2}+C_{n-3}$$E10

and $\left\{S_n\right\}=S_0,S_1,S_2,S_3,\dots .,S_n$where,

$$\begin{array}{c}{S}_0=C_6-C_5-C_4-C_3-C_2-C_1-C_0\\ S_1=C_7-C_6-C_5-C_4-C_3-C_2-C_1\\ S_2=C_8-C_7-C_6-C_5-C_4-C_3-C_2\\ S_3=C_9-C_8-C_7-C_6-C_5-C_4-C_3\\ S_4=C_{10}-C_9-C_8-C_7-C_6-C_5-C_4\end{array}$$E11

$$\text{with},S_n=C_{n-1}+C_{n-2},n\ge 2$$E12

From (4) and (6) we have for $n\ge 11$

$$P_n=\begin{array}{l}{C}_{n-2}+C_{n-3}+C_{n-4}+C_{n-5}+C_{n-6}+C_{n-7}+C_{n-3}+C_{n-4}{C}_{n-5}+C_{n-6}\\ +C_{n-7}+C_{n-8}+C_{n-4}+C_{n-5}+C_{n-6}+C_{n-7}+C_{n-8}+C_{n-9}+C_{n-5}+C_{n-6}\\ +C_{n-7}+C_{n-8}+C_{n-9}+C_{n-10}+C_{n-6}+C_{n-7}+C_{n-8}+C_{n-9}+C_{n-10}+C_{n-11}\end{array}$$

$$P_n=P_{n-1}+P_{n-2}+P_{n-3}+P_{n-4}+P_{n-5}+P_{n-6}$$

Now, from Eqs. (5) and (6),

$$P_{10}=\begin{array}{l}\left(C_8+C_7+C_6+C_5+C_4\right)+\left(C_7+C_6+C_5+C_4+C_3\right)\\ +\left(C_6+C_5+C_4+C_3+C_2\right)+\left(C_5+C_4+C_3+C_2+C_1\right)\\ +\left(C_4+C_3+C_2+C_1+C_0\right)+\left(C_7-C_6-C_5-C_4\right)\end{array}$$

$$P_{10}=P_9+P_8+P_7+P_6+P_5+P_4$$

$$\text{Similarly},P_9=P_8+P_7+P_6+P_5+P_4+P_3$$

$$P_8=P_7+P_6+P_5+P_4+P_3+P_2$$

$$P_7=P_6+P_5+P_4+P_3+P_2+P_1$$

Hence, we have for $n\ge 6$

$$P_n=P_{n-1}+P_{n-2}+P_{n-3}+P_{n-4}+P_{n-5}+P_{n-6}$$E13

Proceeding on similar lines, it can be shown that for $n\ge 6$.

$$\begin{array}{c}{Q}_n=C_{n-2}+C_{n-3}+C_{n-4}+C_{n-5}+C_{n-6}+C_{n-7}\\ +C_{n-3}+C_{n-4}+C_{n-5}+C_{n-6}+C_{n-7}+C_{n-8}\\ +C_{n-4}+C_{n-5}+C_{n-6}+C_{n-7}+C_{n-8}+C_{n-9}\\ +C_{n-5}+C_{n-6}+C_{n-7}+C_{n-8}+C_{n-9}+C_{n-10}\\ Q_n=Q_{n-1}+Q_{n-2}+Q_{n-3}+Q_{n-4}+Q_{n-5}+Q_{n-6},n\ge 6\end{array}$$E14

Proceeding on similar lines it can be shown that for $n\ge 6$

$$\begin{array}{c}{R}_n=C_{n-2}+C_{n-3}+C_{n-4}+C_{n-5}+C_{n-6}+C_{n-7}\\ +C_{n-3}+C_{n-4}+C_{n-5}+C_{n-6}+C_{n-7}+C_{n-8}\\ +C_{n-4}+C_{n-5}+C_{n-6}+C_{n-7}+C_{n-8}+C_{n-9}\\ R_n=R_{n-1}+R_{n-2}+R_{n-3}+R_{n-4}+R_{n-5}+R_{n-6},n\ge 6\end{array}$$E15

Proceeding on similar lines it can be shown that for $n\ge 6$

$$\begin{array}{c}{S}_n=C_{n-2}+C_{n-3}+C_{n-4}+C_{n-5}+C_{n-6}+C_{n-7}\\ +C_{n-3}+C_{n-4}+C_{n-5}+C_{n-6}+C_{n-7}+C_{n-8},n\ge 6\\ S_n=S_{n-1}+S_{n-2}+S_{n-3}+S_{n-4}+S_{n-5}+S_{n-6},n\ge 6\end{array}$$E16

Thus, the four sequences {$P_n$}, {$Q_n$}, {$R_n$}, and {$S_n$} are special cases of sequence {$C_n$} and all obtained by taking different initial values [11, 12].

On taking,

$$\begin{array}{c}{C}_0=C_1=C_2=0,C_3=C_4=1,C_5=2\\ C_0=C_1=0,C_2=1,C_3=0,C_4=1,C_5=2\\ C_0=0,C_1=1,C_2=C_3=0,C_4=1,C_5=2\\ C_0=1,C_1=C_2=C_3=0,C_4=1,C_5=2\\ C_0=C_1=C_2=C_3=0,C_4=1,C_5=2\end{array}$$E17

$$\mathrm{0,0,0,1,1,2,4,8,16,32,63},\dots J_n,\dots$$

$$\mathrm{0,0,1,0,1,2,4,8,16,31,62},\dots K_n,\dots$$

$$\mathrm{0,1,0,0,1,2,4,8,15,30,59},\dots L_n,\dots$$

$$\mathrm{1,0,0,0,1,2,4,7,14,28,56},\dots M_n,\dots$$

$$\mathrm{0,0,0,0,1,2,3,6,12,24,48},\dots N_n,\dots$$

Here, we find that

$$K_n=J_{n-1}+J_{n-2}+J_{n-3}+J_{n-4}+J_{n-5}$$

$$L_n=J_{n-1}+J_{n-2}+J_{n-3}+J_{n-4}$$

$$M_n=J_{n-1}+J_{n-2}+J_{n-3}$$

$$N_n=J_{n-1}+J_{n-2}$$

Hence, we say that {$J_n$} is $C_n$ type sequence, while {$K_n$} is $P_n$ type sequence, and {$L_n$} is $Q_n$ type sequence, while {$M_n$} is$R_n$ type sequence, and {$N_n$} is $S_n$ type sequence.

2.1 Linear sums and some properties

We have derived simple properties [2, 13, 14] of the sequences {$C_n$}, {$P_n$}, {$Q_n$}, {$R_n$}, and{$S_n$}, expressing each of the terms $C_6,C_7,C_8,\dots .,C_{n+5}$C₆, as the sum of its six preceding terms, as given in (4) adding both sides we obtained on

Simplification:

$$\sum _{i=0}^n{C}_i=\frac{1}{5}\left\{C_{n+5}-C_{n+3}-2C_{n+2}-3C_{n+1}+C_n-\left(C_5-C_3-2C_2-3C_1-4C_0\right)\right\}$$E18

On using (4), (5), (7), (9), and (12), we get

$$\sum _{i=0}^n{C}_{6i}=\sum _{i=0}^{6n-1}{C}_i+C_0$$E19

$$\sum _{i=0}^n{C}_{6i+2}=\sum _{i=0}^{6n+1}{C}_i+P_0$$E20

$$\sum _{i=0}^n{C}_{6i+3}=\sum _{i=0}^{6n+2}{C}_i+Q_0$$E21

$$\sum _{i=0}^n{C}_{6i+4}=\sum _{i=0}^{6n+3}{C}_i+R_0$$E22

$$\sum _{i=0}^n{C}_{6i+5}=\sum _{i=0}^{6n+4}{C}_i+S_0$$E23

$$\sum _{i=0}^n{C}_{6i+6}=\sum _{i=0}^{6n+5}{C}_i+\left(S_1-C_0\right)$$E24

$$\sum _{i=0}^n{C}_{6i+5}=\sum _{i=0}^{6n+4}{C}_i+\left(R_1-C_0\right)$$E25

$$\sum _{i=0}^n{C}_{6i+4}=\sum _{i=0}^{6n+3}{C}_i+\left(Q_1-C_0\right)$$E26

$$\sum _{i=0}^n{C}_{6i+3}=\sum _{i=0}^{6n+2}{C}_i\left(P_1-C_0\right)$$E27

2.2 Property of sequence {${\mathit{J}}_{\mathit{n}-\mathbf{2}}$}

Theorem: For the sequence {$J_n$} we have,

$$\left|\begin{array}{cc}\begin{array}{ccc}{J}_n& J_{n+1}& J_{n+2}\\ J_{n+1}& J_{n+2}& J_{n+3}\\ J_{n+2}& J_{n+3}& J_{n+4}\end{array}& \begin{array}{ccc}{J}_{n+3}& J_{n+4}& J_{n+5}\\ J_{n+4}& J_{n+5}& J_{n+6}\\ J_{n+5}& J_{n+6}& J_{n+7}\end{array}\\ \begin{array}{ccc}{J}_{n+3}& J_{n+4}& J_{n+5}\\ J_{n+4}& J_{n+5}& J_{n+6}\\ J_{n+5}& J_{n+6}& J_{n+7}\end{array}& \begin{array}{ccc}{J}_{n+6}& J_{n+7}& J_{n+8}\\ J_{n+7}& J_{n+8}& J_{n+9}\\ J_{n+8}& J_{n+9}& J_{n+10}\end{array}\end{array}\right|={\left(-1\right)}^{n+1}$$E28

Proof: Consider the determinant –

$$∆=\left|\begin{array}{cc}\begin{array}{ccc}1& 1& 1\\ 1& 0& 0\\ 0& 1& 0\end{array}& \begin{array}{ccc}1& 1& 1\\ 0& 0& 0\\ 0& 0& 0\end{array}\\ \begin{array}{ccc}0& 0& 1\\ 0& 0& 0\\ 0& 0& 0\end{array}& \begin{array}{ccc}0& 0& 0\\ 1& 0& 0\\ 0& 1& 0\end{array}\end{array}\right|$$

The value of this determinant is 1, we have

$${∆}^2=\left|\begin{array}{cc}\begin{array}{ccc}2& 2& 2\\ 1& 1& 1\\ 1& 0& 0\end{array}& \begin{array}{ccc}2& 2& 2\\ 1& 1& 1\\ 0& 0& 0\end{array}\\ \begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}& \begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 1& 0& 0\end{array}\end{array}\right|$$

Now, by mathematical induction,

$${∆}^n=\left|\begin{array}{cc}\begin{array}{ccc}{J}_{n+1}& K_{n+1}& L_{n+1}\\ J_n& K_n& L_n\\ J_{n-1}& K_{n-1}& L_{n-1}\end{array}& \begin{array}{ccc}{M}_{n+1}& N_{n+1}& J_n\\ M_n& N_n& J_{n-1}\\ M_{n-1}& N_{n-1}& J_{n-2}\end{array}\\ \begin{array}{ccc}{J}_{n-2}& K_{n-2}& L_{n-2}\\ J_{n-3}& K_{n-3}& L_{n-3}\\ J_{n-4}& K_{n-4}& L_{n-4}\end{array}& \begin{array}{ccc}{M}_{n-2}& N_{n-2}& J_{n-3}\\ M_{n-3}& N_{n-3}& J_{n-4}\\ M_{n-4}& N_{n-4}& J_{n-5}\end{array}\end{array}\right|$$

Now, writing $M_{n+1}=J_n+J_{n-1}$ the R.H.S. can be written as the sum of two determinants, one of which is zero, Therefore,

$${∆}^n=\left|\begin{array}{cc}\begin{array}{ccc}{J}_{n+1}& K_{n+1}& L_{n+1}\\ J_n& K_n& L_n\\ J_{n-1}& K_{n-1}& L_{n-1}\end{array}& \begin{array}{ccc}{M}_{n+1}& J_{n-1}& J_n\\ M_n& J_{n-2}& J_{n-1}\\ M_{n-1}& J_{n-3}& J_{n-2}\end{array}\\ \begin{array}{ccc}{J}_{n-2}& K_{n-2}& L_{n-2}\\ J_{n-3}& K_{n-3}& L_{n-3}\\ J_{n-4}& K_{n-4}& L_{n-4}\end{array}& \begin{array}{ccc}{M}_{n-2}& J_{n-4}& J_{n-3}\\ M_{n-3}& J_{n-5}& J_{n-4}\\ M_{n-4}& J_{n-6}& J_{n-5}\end{array}\end{array}\right|$$

Now, writing $M_{n+1}=J_n+J_{n-1}+J_{n-2}$, the R.H.S. can be written as the sum of three determinants, two of which are zero. Therefore,

$${∆}^n=\left|\begin{array}{cc}\begin{array}{ccc}{J}_{n+1}& K_{n+1}& L_{n+1}\\ J_n& K_n& L_n\\ J_{n-1}& K_{n-1}& L_{n-1}\end{array}& \begin{array}{ccc}{J}_{n-2}& J_{n-1}& J_n\\ J_{n-3}& J_{n-2}& J_{n-1}\\ J_{n-4}& J_{n-3}& J_{n-2}\end{array}\\ \begin{array}{ccc}{J}_{n-2}& K_{n-2}& L_{n-2}\\ J_{n-3}& K_{n-3}& L_{n-3}\\ J_{n-4}& K_{n-4}& L_{n-4}\end{array}& \begin{array}{ccc}{J}_{n-5}& J_{n-4}& J_{n-3}\\ J_{n-6}& J_{n-5}& J_{n-4}\\ J_{n-7}& J_{n-6}& J_{n-5}\end{array}\end{array}\right|$$

Now, writing $L_{n+1}=J_n+J_{n-1}+J_{n-2}+J_{n-3}$L_n + 1, the R.H.S. can be written as the sum of four determinants, three of which are zero. Therefore,

$${∆}^n=\left|\begin{array}{cc}\begin{array}{ccc}{J}_{n+1}& K_{n+1}& J_{n-3}\\ J_n& K_n& J_{n-4}\\ J_{n-1}& K_{n-1}& J_{n-5}\end{array}& \begin{array}{ccc}{J}_{n-2}& J_{n-1}& J_n\\ J_{n-3}& J_{n-2}& J_{n-1}\\ J_{n-4}& J_{n-3}& J_{n-2}\end{array}\\ \begin{array}{ccc}{J}_{n-2}& K_{n-2}& J_{n-6}\\ J_{n-3}& K_{n-3}& J_{n-7}\\ J_{n-4}& K_{n-4}& J_{n-8}\end{array}& \begin{array}{ccc}{J}_{n-5}& J_{n-4}& J_{n-3}\\ J_{n-6}& J_{n-5}& J_{n-4}\\ J_{n-7}& J_{n-6}& J_{n-5}\end{array}\end{array}\right|$$

Now, writing $K_{n+1}=J_n+J_{n-1}+J_{n-2}+J_{n-3}+J_{n-4}$ the R.H.S. can be written as the sum of five determinants, four of which are zero. Therefore,

$${∆}^n=\left|\begin{array}{cc}\begin{array}{ccc}{J}_{n+1}& J_{n-4}& J_{n-3}\\ J_n& J_{n-5}& J_{n-4}\\ J_{n-1}& J_{n-6}& J_{n-5}\end{array}& \begin{array}{ccc}{J}_{n-2}& J_{n-1}& J_n\\ J_{n-3}& J_{n-2}& J_{n-1}\\ J_{n-4}& J_{n-3}& J_{n-2}\end{array}\\ \begin{array}{ccc}{J}_{n-2}& J_{n-7}& J_{n-6}\\ J_{n-3}& J_{n-8}& J_{n-7}\\ J_{n-4}& J_{n-8}& J_{n-8}\end{array}& \begin{array}{ccc}{J}_{n-5}& J_{n-4}& J_{n-3}\\ J_{n-6}& J_{n-5}& J_{n-4}\\ J_{n-7}& J_{n-6}& J_{n-5}\end{array}\end{array}\right|$$

On arranging, we get

$${∆}^n=\left|\begin{array}{cc}\begin{array}{ccc}{J}_{n+1}& J_n& J_{n-1}\\ J_n& J_{n-1}& J_{n-2}\\ J_{n-1}& J_{n-2}& J_{n-3}\end{array}& \begin{array}{ccc}{J}_{n-2}& J_{n-3}& J_{n-4}\\ J_{n-3}& J_{n-4}& J_{n-5}\\ J_{n-4}& J_{n-5}& J_{n-6}\end{array}\\ \begin{array}{ccc}{J}_{n-2}& J_{n-3}& J_{n-4}\\ J_{n-3}& J_{n-4}& J_{n-5}\\ J_{n-4}& J_{n-5}& J_{n-6}\end{array}& \begin{array}{ccc}{J}_{n-5}& J_{n-6}& J_{n-7}\\ J_{n-6}& J_{n-7}& J_{n-8}\\ J_{n-7}& J_{n-8}& J_{n-9}\end{array}\end{array}\right|$$

Putting, n-9 = m or n = m + 9 and substituting all the Δ’s, we obtain,

$${\left(-1\right)}^{m+9}=\left|\begin{array}{cc}\begin{array}{ccc}{J}_{m+10}& J_{m+9}& J_{m+8}\\ J_{m+9}& J_{m+8}& J_{m+7}\\ J_{m+8}& J_{m+7}& J_{m+6}\end{array}& \begin{array}{ccc}{J}_{m+7}& J_{m+6}& J_{m+5}\\ J_{m+6}& J_{m+5}& J_{m+4}\\ J_{m+5}& J_{m+4}& J_{m+3}\end{array}\\ \begin{array}{ccc}{J}_{m+7}& J_{m+6}& J_{m+5}\\ J_{m+6}& J_{m+5}& J_{m+4}\\ J_{m+5}& J_{m+4}& J_{m+3}\end{array}& \begin{array}{ccc}{J}_{m+4}& J_{m+3}& J_{m+2}\\ J_{m+3}& J_{m+2}& J_{m+1}\\ J_{m+2}& J_{m+1}& J_m\end{array}\end{array}\right|$$

Rearranging the determinant and replacing m with n we get the required result (28)

2.3 Generating matrix {Cn}

In this section, we will obtain some identities with the help of generating matrix, we consider the matrix,

$$\left[T\right]=\left|\begin{array}{cc}\begin{array}{ccc}1& 1& 1\\ 1& 0& 0\\ 0& 1& 0\end{array}& \begin{array}{ccc}1& 1& 1\\ 0& 0& 0\\ 0& 0& 0\end{array}\\ \begin{array}{ccc}0& 0& 1\\ 0& 0& 0\\ 0& 0& 0\end{array}& \begin{array}{ccc}0& 0& 0\\ 1& 0& 0\\ 0& 1& 0\end{array}\end{array}\right|$$E29

By mathematical induction, we can show that:

$${\left[T\right]}^n=\left|\begin{array}{cc}\begin{array}{ccc}{J}_{n+1}& K_{n+1}& L_{n+1}\\ J_n& K_n& L_n\\ J_{n-1}& K_{n-1}& L_{n-1}\end{array}& \begin{array}{ccc}{M}_{n+1}& N_{n-1}& J_n\\ M_n& N_{n-2}& J_{n-1}\\ M_{n-1}& N_{n-3}& J_{n-2}\end{array}\\ \begin{array}{ccc}{J}_{n-2}& K_{n-2}& L_{n-2}\\ J_{n-3}& K_{n-3}& L_{n-3}\\ J_{n-4}& K_{n-4}& L_{n-4}\end{array}& \begin{array}{ccc}{M}_{n-2}& N_{n-4}& J_{n-3}\\ M_{n-3}& N_{n-5}& J_{n-4}\\ M_{n-4}& N_{n-6}& J_{n-5}\end{array}\end{array}\right|$$E30

where $n\ge 5$

and

$$\left[C_n,C_{n-1},C_{n-2},C_{n-3},C_{n-4},C_{n-5}\right]={\left[T\right]}^n\left[C_5,C_4,C_3,C_2,C_1,C_0\right],n\ge 5$$E31

On using (30) and (31), we get:

$$\left|\begin{array}{c}\begin{array}{c}{C}_{n+P}\\ C_{n+P-1}\\ C_{n+P-2}\end{array}\\ \begin{array}{c}{C}_{n+P-3}\\ C_{n+P-4}\\ C_{n+P-5}\end{array}\end{array}\right|=\left|\begin{array}{cc}\begin{array}{ccc}{J}_{n+1}& K_{n+1}& L_{n+1}\\ J_n& K_n& L_n\\ J_{n-1}& K_{n-1}& L_{n-1}\end{array}& \begin{array}{ccc}{M}_{n+1}& N_{n-1}& J_n\\ M_n& N_{n-2}& J_{n-1}\\ M_{n-1}& N_{n-3}& J_{n-2}\end{array}\\ \begin{array}{ccc}{J}_{n-2}& K_{n-2}& L_{n-2}\\ J_{n-3}& K_{n-3}& L_{n-3}\\ J_{n-4}& K_{n-4}& L_{n-4}\end{array}& \begin{array}{ccc}{M}_{n-2}& N_{n-4}& J_{n-3}\\ M_{n-3}& N_{n-5}& J_{n-4}\\ M_{n-4}& N_{n-6}& J_{n-5}\end{array}\end{array}\right|\left|\begin{array}{c}\begin{array}{c}{C}_n\\ C_{n-1}\\ C_{n-2}\end{array}\\ \begin{array}{c}{C}_{n-3}\\ C_{n-4}\\ C_{n-5}\end{array}\end{array}\right|$$

From this we obtain:

$$C_{n+P}=J_{P+1}{D}_n+K_{P+1}{D}_{n-1}+L_{P+1}{D}_{n-2}+M_{P+1}{D}_{n-3}+N_{P+1}{D}_{n-4}+J_{P+1}{D}_{n-5}$$E32

Let us now consider the matrix [W], which is the transpose of the matrix [T] in,

$$\left[W\right]=\left[T\right]=\left|\begin{array}{cc}\begin{array}{ccc}1& 1& 0\\ 1& 0& 1\\ 1& 0& 0\end{array}& \begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 1& 0& 0\end{array}\\ \begin{array}{ccc}1& 0& 0\\ 1& 0& 0\\ 1& 0& 0\end{array}& \begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\end{array}\right|$$

It can be shown that the sequence

$$C_4,P_5,Q_5,,R_5,S_5,C_5\dots .,C_{n-1},P_n,Q_n,,R_n,S_n,C_n$$E33

is generated by matrix [W]

$$\left[C_n,P_n,Q_n,R_n,S_n,C_{n-1}\right]={\left[W\right]}^{n-5}\left[C_5,P_5,Q_5,R_5,S_5,C_4\right],n\ge 5$$E34

On using (33) and (34), we get

$$\begin{array}{c}\left[C_{n+P},P_{n+P},Q_{n+P},R_{n+P},S_{n+P},C_{n+P}\right]\\ ={\left[W\right]}^P\left[C_n,P_n,Q_n,R_n,S_n,C_{n-1}\right]\\ =\left|\begin{array}{cc}\begin{array}{ccc}{J}_{P+1}& J_P& J_{P-1}\\ K_{P+1}& K_P& K_{P-1}\\ L_{P+1}& L_P& L_{P-1}\end{array}& \begin{array}{ccc}{J}_{P-2}& J_{P-3}& J_{P-4}\\ K_{P-2}& K_{P-3}& K_{P-4}\\ L_{P-2}& L_{P-3}& L_{P-4}\end{array}\\ \begin{array}{ccc}{M}_{P+1}& M_P& M_{P-1}\\ N_{P+1}& N_P& N_{P-1}\\ J_P& J_{P-1}& J_{P-2}\end{array}& \begin{array}{ccc}{M}_{P-2}& M_{P-3}& M_{P-4}\\ N_{P-2}& N_{P-3}& N_{P-4}\\ J_{P-3}& J_{P-4}& J_{P-5}\end{array}\end{array}\right|\left|\begin{array}{c}\begin{array}{c}{C}_n\\ P_n\\ Q_n\end{array}\\ \begin{array}{c}{R}_n\\ S_n\\ C_{n-1}\end{array}\end{array}\right|\end{array}$$

$$C_{n+P}=J_{P+1}{C}_n+J_P{P}_n+J_{P-1}{Q}_n+J_{P-2}{R}_n+J_{P-3}{S}_n+J_{P-4}{C}_{n-1}$$

$$P_{n+P}=K_{P+1}{C}_n+K_P{P}_n+K_{P-1}{Q}_n+K_{P-2}{R}_n+K_{P-3}{S}_n+K_{P-4}{C}_{n-1}$$

$$Q_{n+P}=L_{P+1}{C}_n+L_P{P}_n+L_{P-1}{Q}_n+L_{P-2}{R}_n+L_{P-3}{S}_n+L_{P-4}{C}_{n-1}$$

$$R_{n+P}=M_{P+1}{C}_n+M_P{P}_n+M_{P-1}{Q}_n+M_{P-2}{R}_n+M_{P-3}{S}_n+M_{P-4}{C}_{n-1}$$

$$S_{n+P}=N_{P+1}{C}_n+N_P{P}_n+N_{P-1}{Q}_n+N_{P-2}{R}_n+N_{P-3}{S}_n+N_{P-4}{C}_{n-1}$$

Application:

We can introduce generalized Fibonacci n-step polynomials. Based on generalized Fibonacci n-step polynomials, we can define a new class of square matrix of order n and we can state a new coding theory called generalized Fibonacci n-step theory.

3. Discussion

Mathematics has enormous potential for solving the various problems of daily life. The Fibonacci polynomials are a polynomial sequence that can be considered as generalization sequences worked upon by many mathematicians earlier like as Atanassov [11], Harne & Parihar [4], and Georgiev and Atanassov [8] in accordance with our findings. The chapter has wider acceptance for the fruitful study of various case studies as illustrated in the current citation, which is well supported by the earlier studies too.

4. Conclusions

There are many known identities for the Fibonacci recursion relation. We define the sequence {$C_n$} and its four comparison sequences {$P_n$}, {$Q_n$}, {$R_n$}, and {$S_n$}. We drive linear sum properties of comparison sequence. We also derive generating matrix for the sequence {$C_n$}.