Determinantal Identities of-Tribonacci Sequences

Pankaj

Department of Mathematics,

Indira Gandhi University, Meerpur (Rewari)-122502, Haryana, India

*Corresponding Author E-mail: pankajarora1242@yahoo.com

Abstract:

In this paper, we find some new determinantal identities using generalizedk-Tribonacci sequences which are defined as:

where  are positive integers with

The generalized k-Tribonacci sequences are

KEYWORDS:k-Fibonacci sequence, k-Lucas sequence, k-Tribonaccisequence

2010 MATHEMATICS SUBJECT CLASSIFICATIONS: 11Bxx, 11B39, 11B83, 11K31

1. INTRODUCTION:

The well known Fibonacci sequence has many interesting properties. The Fibonacci sequence has been generalized in many ways, some by preserving the initial conditions, and others by preserving the recurrence relation. Many properties of these types of sequences have been derived (See [1], [2], [3], [4], [5], [6], [7], [8], [9]). In this paper, we find some new determinantal identities using generalized k- Tribonacci sequences. Here, we give definitions of k-Fibonacci, k-Lucas and k-Tribonacci sequences.

1.1. Definition: The k-Fibonacci sequence  is defined as,

for with

1.2. Definition: The k-Lucas sequence  is defined as,

for with

1.3. Definition: The k-Tribonacci sequence  is defined as,

for with

2. Generalized -Tribonacci Sequences

Now we define a family of - Tribonacci sequences as

where  are positive integers with

The generalized k-Tribonacci sequences are

                          (1)

(2)

                     (3)

 (4)

3. Determinantal Identities of k-Tribonacci Sequences

3.1. Theorem: If  are positive integers with  then

Proof: Let

Assume then by (1),

Now,                    

Taking  common from  and respectively, we get

Taking  common from  and respectively, we get

Applying, we have

Applying  we have

Applying  we have

Expanding by first row, we get

Expanding by first row, we get

Put  and  we get

The following identities can be proved in a similar way as in Theorem 3.1.

3.2. Theorem: If  are positive integers with  then

3.3. Theorem: If  are positive integers with  then

3.4. Theorem: If  are positive integers with then

3.5. Theorem: If  are positive integers with  then

3.6. Theorem: If  are positive integers with  then

4. REFERENCES:

[1] S. Falcon, On the generating matrices of the k-Fibonacci numbers, Proyecciones Journal of Mathematics, 32 (4) (2013), 347-357.

[2] S. Falcon, and A. Plaza, On the k-Fibonacci numbers, Chaos, Solitons and Fractals, 5(32) (2007), 1615-1624.

[3] S. Falcon, and A. Plaza, The k-Fibonacci hyperbolic functions, Chaos, Solitons and Fractals, 38(2) (2008), 409-420.

[4] S. Falcon, and A.  Plaza, The k-Fibonacci sequence and the Pascal 2-triangle, Chaos, Solitons and Fractals, 33(1) (2007), 38-49.

[5] A. Feng, Fibonacci identities via determinant of tridiagonal matrix, Applied Mathematics and Computation, 217 (2011), 5978-5981.

[6] A. D. Godase, and M. B. Dhakne, Determinantal Identities for k-Lucas Sequence, Research Gate Pub., 2015.

[7] Roji Lather, Ashish and Manoj Kumar, On the Stability of Tribonacci and k- Tribonacci Functional Equations in Modular Space, International Journal of Pure and Applied Mathematics, 104(2) (2015), 265-284.

[8] Roji Lather and Manoj Kumar, Stability of k-Tribonacci Functional Equation in Non-Archimedean Space, International Journal of Computer Applications, 128(14) (2015), 27-30.

[9] Pankaj, Some New Determinantal Identities of -Pell Sequences, Journal of Combinatorics, Information & System Sciences, 41(4) (2016), 207-213.

[10] N. J. Sloane, The online encyclopaedia of integer sequences, (2006).

Received on 27.05.2018      Modified on 28.06.2018

Accepted on 25.07.2018      ©A&V Publications All right reserved

Research J. Science and Tech. 2018; 10(3):197-200.

DOI: 10.5958/2349-2988.2018.00027.X