Keywords: Lucas Polynomials, rising diagonal function, descending diagonal function and generating matrix
Turkish Journal of Analysis and Number Theory, 2015 3 (2),
pp 4952.
DOI: 10.12691/tjant323
Received January 29, 2015; Revised March 10, 2015; Accepted April 19, 2015
Copyright © 2015 Science and Education Publishing. All Rights Reserved.
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 CaseI
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 kLucas 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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