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Diagonal Function of k-Lucas Polynomials

Yogesh Kumar Gupta*, V. H. Badshah, Mamta Singh, Kiran Sisodiya

School of Studies in Mathematics, Vikram University Ujjain, (M. P.) India

Abstract: The Lucas polynomials are famous for possessing wonderful and amazing properties and identities. In this paper, Diagonal function of k-Lucas Polynomials is introduced and defined by Gn+1(x)=kxGn(x)+Gn-2,(x), n≥1. with G₀(x)=2. and G₁(x)=1 Some Lucas Polynomials, rising & descending diagonal function and generating matrix established and derived by standard methods.

Keywords: Lucas Polynomials, rising diagonal function, descending diagonal function and generating matrix

Received January 29, 2015; Revised March 10, 2015; Accepted May 04, 2015

Citation: Yogesh Kumar Gupta, V. H. Badshah, Mamta Singh, Kiran Sisodiya, Diagonal Function of k-Lucas Polynomials. Turkish Journal of Analysis and Number Theory, 3 (2). 49-52. January 2015. doi:10.12691/tjant-3-2-3

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

[picture] | (1.1)

The first few Lucas polynomials are

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

[picture] | (1.2)

[picture] | (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 {Gn} and {Pn}

We have the pair of sequence {Gn} and {Pn} for which,

The first few terms of the sequence {Gn} are

[picture] | (2.1)

The first few terms of the sequence {Pn} are

[picture] | (2.2)

3. Rising Diagonal Function

Consider the rising diagonal function of x, Un (x), un (x) for (2.1) and (2.2) respectively,

[picture] | (3.1)

[picture] | (3.2)

Now, we define

[picture] | (3.3)

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

Theorem (1). If Un (x) and un(x) are rising diagonal functions of x for sequence {Gn} and {Pn} respectively, than for,

[picture] | (3.4)

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

Special Case-I

If Un (x) and un(x) are rising diagonal functions of x f sequence {Gn} and {Pn} respectively, than for n=3, n=4.

[picture] | (3.5)

4. Descending Diagonal Function

From (2.1) and (2.2), the descending diagonal function of x, Qi(x), qi(x) are

[picture] | (4.1)

[picture] | (4.2)

Now, we define

[picture] | (4.3)

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

[picture] | (4.4)

[picture] | (4.5)

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

Theorem (2). If Qn(x) and qn(x) are descending diagonal function of x for Sequence {Gn} and {Pn} respectively, than for.

a)

b)

5. Generating Matrix

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

[picture] | (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 {Gn} we define ,

Proof. For sequence {Gn}, we have

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

[picture] | (5.2)

[picture] | (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 {Gn} we define,

Proof. For sequence {Gn}, we have

If A is any square Matrix, then we know that

[picture] | (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 {Pn} defend in equation (1.1) we consider the matrix

[picture] | (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 {Pn} we define,

Proof. For sequence {Pn}, we have

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

[picture] | (6.2)

By mathematical induction

[picture] | (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 {Pn} we define,

Proof. For sequence {Gn}, we have

If A is any square Matrix, then we know that

[picture] | (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.

References

[1]   | A.F Hordam, Diagonal Function, the Fibonacci Quarterly, Vol. 16, 19-36.
[2]   | Alexandra Lupas Guide of Fibonacci and Lucas polynomials, Octagon Mathematics Magazine Vo. 17, No.1 (1999), 2-12.
[3]   | B.S.Porov, A note on the sum of Fibonacci and Lucas polynomials, The Fill Quarterly 1970, 428-438.
[4]   | D.V. Jaiswal, “Some Metric Properties of a Generalized Fibonacci Sequence’’, Labdev Journal of Science and technology, India vol.11-A, No.1, 1973, 1-3.
[5]   | Jr. V.E. Hoggatt, Fibonacci and Lucas Numbers, Houshton Mifflin Company, Borton, 1965.
[6]   | Koshy, T. Fibonacci and Lucas number with application, Wiley, 2001.
[7]   | M. Catalan, An Identity for Lucas Polynomials, Fibonacci Quarterly Vol 43, No.1 2005.
[8]   | Singh, B., S. Teeth, and Harne Diagonal Function of Fibonacci Polynomials, chh. J. Sci. Tech 2005, 97-102.
[9]   | Vajda, S. Fibonacci and Lucas numbers and the golden section, Ellis Horwood Limited, Chi Chester, England, 1989.
[10]   | Vorobyou, N.N., The Fibonacci number, D.C. health company, Boston, 1963.