﻿ Some Identities of Tribonacci Polynomials

### Some Identities of Tribonacci Polynomials

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

Turkish Journal of Analysis and Number Theory

## Some Identities of Tribonacci Polynomials

Yogesh Kumar Gupta1, , V. H. Badshah1, Mamta Singh2, Kiran Sisodiya1

1School of Studies in Mathematics, Vikram University Ujjain, (M. P.), India

2Department of Mathematical Sciences and Computer applications, Bundelkhand University, Jhansi (U. P.), India

### Abstract

The Tribonacci polynomial is famous for possessing wonderful and amazing properties. Tribonacci polynomials tn(x) defined by the recurrence relation tn+3(x)=x2tn+2(x)+xtn+1(x)+tn(x) for n0 with to(x) =o, t1(x)=1, t2(x)=x2. In this paper, we introduce some identities Tribonacci polynomials by standard techniques.

### Cite this article:

• Yogesh Kumar Gupta, V. H. Badshah, Mamta Singh, Kiran Sisodiya. Some Identities of Tribonacci Polynomials. Turkish Journal of Analysis and Number Theory. Vol. 4, No. 1, 2016, pp 20-22. http://pubs.sciepub.com/tjant/4/1/4
• Gupta, Yogesh Kumar, et al. "Some Identities of Tribonacci Polynomials." Turkish Journal of Analysis and Number Theory 4.1 (2016): 20-22.
• Gupta, Y. K. , Badshah, V. H. , Singh, M. , & Sisodiya, K. (2016). Some Identities of Tribonacci Polynomials. Turkish Journal of Analysis and Number Theory, 4(1), 20-22.
• Gupta, Yogesh Kumar, V. H. Badshah, Mamta Singh, and Kiran Sisodiya. "Some Identities of Tribonacci Polynomials." Turkish Journal of Analysis and Number Theory 4, no. 1 (2016): 20-22.

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### 1. Introduction

Mathematics can be considered as the underlying order of the universe, and the Fibonacci numbers is one of the most fascinating discovery made in the mathematical world. Among numerical sequences, the Fibonacci sequence has achieved a kind of celebrity status and has been studied extensively in number theory, applied mathematics, physics, computer science, and biology . The Fibonacci numbers are famous for possessing wonderful and amazing properties. A similar interpretation also exists for Lucas sequence. The Fibonacci numbers have been studied both for their applications and the mathematical beauty of rich and interesting identities that they satisfy.

The Fibonacci sequence {fn} of number Fn is defined by the recurrence relation for , (1.1)

Binet Formula for Fibonacci number is defined by (1.2)

In 1883, catalan, E  was considered related set of polynomials which satisfies the recurrence relations. The name Fibonacci polynomials is also given to the solution of the relation (1.3)

With , investigated by Jacasthal, E . Byrd, p.F  defined the Fibonacci polynomials by the recurrence relation. (1.4)

with initial conditions swamy, M.N.S  and Hoggatt, V.E.  almost simultaneously defined the Fibonacci polynomials by (1.5)

with .

Here if we put x=1 in we get which if Fibonacci sequence.

Generating function of Fibonacci Polynomials is defined by Hyper geometric form of generating function of Fibonacci polynomials is Tribonacci number Tn  defined by (1.6)

The Tribonacci numbers are (0, 1, 1, 2, 4, 7, 13, 24…).

The tribonacci polynomial tn(x)  satisfies the following recurrence relation: and to(x)=o, t1(x)=1 t2(x)=x2.

In this chapter we present some identities of Tribonacci Polynomials by standard methods.

### 2. Tribonacci Polynomials

The Tribonacci polynomial tn(x)  satisfies the following recurrence relation: and to(x)=o, t1(x)=1, t2(x)=x2

The first few Tribonacci polynomials are as follows.

t1(x) = 1

t2(x) = x2

t3(x) = x4 +x

t4(x) = x6 +2x3+1

t5(x) = x8 +3x5+3x2

t6(x) = x10 +4x7+6x4 +2x

t7(x) = x12 +5x9+10x4 +7x3+1

t8(x) = x14 +6x11+15x8 +16x5+6x2…and so on.

Obviously, tn (1) is Just the classical Tribonacci number Tn several basic properties its simple form the Tribonacci polynomials have many interesting properties.

The Generating function of Tribonacci polynomials is defined by (2.1)

Now we will define the Hypergeometric form of generating function of Tribonacci polynomials. In this chapter we shall use following results.

(1) (2) (3) (4) ### 3. Some Identities of Tribonacci Polynomials

Now we state and prove some Identities of Tribonacci polynomials theorem.

Theorem (3.1). Proof. we know that the generating function of Tribonacci polynomials is   Theorem (3.2). recurrence relations by the generating function of Tribonacci polynomials we can easily get following recurrence relations. and Proof. The generating function of Tribonacci polynomials is Differentiating (2.1) w.r. to we get   Equating the coefficient of tn+1and tn, we get and Theorem (3.3). Prove that and .

Proof.

The Generating function of Tribonacci Polynomials Put x=0 in eqn (2.1) we get Equating the coefficients of and on both sides respectively, we get Theorem (3.4).Prove that Proof . Differentiating equation (2.1) w.r.t ‘x', we get  Put x=o, we get  on both side respectively, we get ### 4. Conclusion

In this paper, we introduce some identities Tribonacci polynomials by standard techniques. Some basic identities are obtained by method of generating function. Also some identities are obtained in hyper geometric form.

### Acknowledgement

The authors would like to thanks the anonymous referee for carefully reading the paper and for their comments.

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