1. Introduction

The sequence 1,1,2,3,5..., got its name as name as Fibonacci sequence by the Famous Mathematics Francois Edouard Lucas in 1876 ^[7].

Lucas also discovered a new Fibonacci like sequence with different initial condition call it, Lucas Sequence

with initial condition ,

In 1965 Hoggatt, V.E. ^[5] has defined Lucas polynomials by recurrence relation. where

(1.1)

The first few Lucas polynomials are

In this paper, we are using the pair of sequence {G_n} and {P_n} for which,

(1.2)

(1.3)

where k is any positive integer. k= 0, 1, 2, 3...

Using the equation (1.1) and (1.2) and made a rising diagonal function and descending Diagonal Functions.

2. Sequence {G_n} and {P_n}

We have the pair of sequence {G_n} and {P_n} for which,

The first few terms of the sequence {G_n} are

(2.1)

The first few terms of the sequence {P_n} are

(2.2)

3. Rising Diagonal Function

Consider the rising diagonal function of x, U_n (x), u_n (x) for (2.1) and (2.2) respectively,

(3.1)

(3.2)

Now, we define

(3.3)

from equation (3.1), (3.2) and (3.3) we get the following theorem:

Theorem (1). If U_n (x) and u_n(x) are rising diagonal functions of x for sequence {G_n} and {P_n} respectively, than for,

(3.4)

Proof can be obtained by PMI’s method so it is obvious.

Special Case-I

If U_n (x) and u_n(x) are rising diagonal functions of x f sequence {G_n} and {P_n} respectively, than for n=3, n=4.

(3.5)

4. Descending Diagonal Function

From (2.1) and (2.2), the descending diagonal function of x, Q_i(x), q_i(x) are

(4.1)

(4.2)

Now, we define

(4.3)

from (4.1), (4.2) and (4.3) we get for.

(4.4)

(4.5)

from (4.4) and (4.5) we get the following theorem:

Theorem (2). If Q_n(x) and q_n(x) are descending diagonal function of x for Sequence {G_n} and {P_n} respectively, than for.

a)

b)

5. Generating Matrix

For the sequence {G_n} defend in equation (1.1) we consider the matrix

(5.1)

Since, the elements of this matrix are the member of the sequence of Fibonacci Polynomials. We call this matrix as Fibonacci matrix.

Theorem (3). For sequence {G_n} we define ,

Proof. For sequence {G_n}, we have

Since, determinant of matrix A is -1, therefore,

(5.2)

(5.3)

Form equation (3.5.2) and (3.5.3), we get

Since

After multiplying the matrices and equating the corresponding elements, we get

Theorem (4). For sequence {G_n} we define,

Proof. For sequence {G_n}, we have

If A is any square Matrix, then we know that

(5.4)

Where I is identity matrix from equation (5.4) we get

By Mathematical induction, we have

Since

After multiplying the matrices and equating the corresponding elements, we get

6. Generating Matrix

For the sequence {P_n} defend in equation (1.1) we consider the matrix

(6.1)

since, the elements of this matrix are the members of the sequence of Fibonacci polynomials. We call this matrix as Fibonacci Matrix.

Theorem (5). For sequence {P_n} we define,

Proof. For sequence {P_n}, we have

Since, determinant of matrix A is -1, there for,

(6.2)

By mathematical induction

(6.3)

Form equation (3.6.2) and (3.6.3), we get

Since

After multiplying the matrices and equating the corresponding elements, we get

Theorem (6). For sequence {P_n} we define,

Proof. For sequence {G_n}, we have

If A is any square Matrix, then we know that

(6.4)

Where I is identity matrix from equation (5.4) we get

By mathematical indication, we have

Since

After multiplying the matrices and equating the corresponding elements, we get

7. Conclusions

In this paper Diagonal function k-Lucas Polynomials. Some basic rising diagonal function and descending diagonal function and generating matrix derived by standard method.

Acknowledgement

We would like to thank the anonymous referees for numerous helpful suggestions.

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