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Incomplete h(x)-B-Tribonacci Polynomials

S. Arolkar*, Y.S. Valaulikar

Department of Mathematics, D.M.’s College and Research Centre, Assagao-Bardez Goa, India, 403 507

Abstract: In this paper we introduce incomplete h(x)-B-Tribonacci polynomials and obtain recurrence relations satisfied by a new class of polynomials.

Keywords: Incomplete Fibonacci Polynomials, h(x)-B-Tribonacci Polynomials and Incomplete h(x)-B- Tribonacci Polynomials

Citation: S. Arolkar, Y.S. Valaulikar, Incomplete h(x)-B-Tribonacci Polynomials. Turkish Journal of Analysis and Number Theory, 4 (6). 155-158. January 2016. doi:10.12691/tjant-4-6-1

1. Introduction

Fibonacci sequences and polynomials have many interesting properties and they provide wide opportunities to construct more fascinating properties of their own and their extensions see, for example, [3,4,8,10,11,12,13,14]. The B-Tribonacci sequence has been introduced in [1]. Incomplete Fibonacci and Lucas numbers and their properties have been studied in [5]. For further information about the incomplete generalized Fibonacci, Lucas and Tribonacci polynomials see, [6,7,9]. In this paper we now introduce incomplete h(x)-B-Tribonacci polynomials which are a natural extension of h(x)-Fibonacci polynomials earlier given by [6].

Let be a polynomial with real coefficients. The -B- Tribonacci polynomials are defined by

[picture] | (1.1)

where the coefficients on the right hand side are the terms of binomial expansion of and is the nth polynomial. It is also given by

[picture] | (1.2)

where isto the r falling factorial and is the largest integer not greater than

Few polynomials of (1.1) are

and

The generating function of (1.1) is given by

[picture] | (1.3)

For further properties of (1.1), see [2]. We define the extension of incomplete h(x)-Fibonacci polynomials defined in [6] and call it as the incomplete h(x) - B- Tribonacci polynomials.

The incomplete h(x)-B-Tribonacci polynomials are defined by

and

[picture] | (1.4)

Note that if then

2. Some Recurrence Properties of the Polynomials (tB)lh,n(x)

In this section, we obtain some recurrence relations for and some identities for a new class of polynomials. For simplicity we use and

Proposition (2.1). For the recurrence relation of the incomplete h(x)-B- Tribonacci polynomials is

[picture] | (2.1)

Using (1.4), Equ. (2.1) can be rewritten in terms of non-homogeneous recurrence relation as

[picture] | (2.2)

Proof. This is proved by using (2.1) as follows.

We now obtain some identities involving sums.

Proposition (2.2). For

[picture] | (2.3)

Proof. Follows from method of mathematical induction.

Proposition (2.3). For we have

[picture] | (2.4)

Proof. The proof follows by using mathematical induction.

Lemma (2.4). For we have

[picture] | (2.5)

Proof. From (1.3), we have

Differentiating the above equation with respect to h we get,

Comparing both sides the coefficient of we have

That is

[picture] | (2.6)

Also differentiating (1.2) on both sides with respect to h, we get

Thus, we have

This completes the proof.

Proposition (2.5). For we have

Proof. From (1.4), the sum,

In view of (2.5), we complete the proof with the following application.

3. Conclusion

In this paper, we have defined incomplete h(x)-B-Tribonacci polynomials and obtained some identities related to these polynomials.

Acknowledgement

Authors would like to thank the referee for his valuable suggestions in improving the paper.

References

[1]   | Arolkar S. and Valaulikar Y. S., On an Extension of Fibonacci sequences, Bulletin of Marathwada Mathematical Society, 17 (1) (2016), 1-8.
[2]   | Arolkar S. and Valaulikar Y.S., h(x)-B-Tribonacci and h(x)-B- Tri Lucas polynomials. To appear in Kyungpook Mathematical Journal.
[3]   | Falcon S., Plaza A., On the Fibonacci k- numbers, Chaos, Solitons & Fractals 32 (2007) 1615-1624.
[4]   | Falcon S., Plaza A., On k-Fibonacci sequences and polynomials and their derivatives, Chaos, Solitons & Fractals 39 (2009) 1005-1019.
[5]   | Filipponi, P., Incomplete Fibonacci and Lucas numbers, Rendiconti del Cicolo Matematico di Palerno, 45(2),37-56, 1996.
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[9]   | Nazmiye Y. and Necati Taskara, Incomplete Tribonacci –Lucas numbers and polynomial, arXiv: 1404.4168v1 [math.NT] 16 April 2014.
[10]   | Panwar Y. and Singh M., Generalized Bivariate Fibonacci- Like Polynomials, International Journal of Pure Mathematics, Volume 1, 2014, 8-13.
[11]   | Vajda S., Fibonacci and Lucas numbers and the Golden section: Theory and Applications,Dover Publications (2008).
[12]   | Wang T. and Zhang W., Some identities involving Fibonacci, Lucas polynomials and their applications, Bull. Math. Soc. Sci. Math. Rourmanie Tome 55 (103 No. 1, 2012, 95-103).
[13]   | Yogesh Kumar Gupta, V.H. Badshah, Mamata Singh, and Kiran Sisodiya, Some Identities of Tribonacci Polynomials, Turkish Journal of Analysis and Number Theory, 2016,Vol.4, No. 1, 20-22.
[14]   | Yi Y., Zhang W., Some identities involving the Fibonacci polynomials, Fibonacci Quarterly, Vol. 40, (2002), 314-318.