Abstract

In this paper, we will recover the generating functions of generalized polynomials of second order. The technic used her is based on the theory of the so called symmetric functions.

2010 Mathematics Subject Classification. Primary 05E05; Secondary 11B39.

1. Introduction

A second order polynomials sequence is of Fibonacci type (Lucas type) if its Binet formula has a structure similar to that for Fibonacci (Lucas) numbers. In the literature, these types of sequences are known as Generalized Fibonacci Polynomial (GFP). They are actually a natural generalization of the sequence, called the Fibonacci polynomial sequence. However, there is no unique generalization of this sequence, one can refer to articles by several authors like André Jeannin ^{7, 10}, Bergum et al. ¹¹ and Florez et al. ^{14, 15}, to see this. In this paper we use the definition another of our suggested. The generalized polynomials of second order sequence is defined by a recurrence sequence

(1.1)

where and are complex numbers.

The study of identities for Fibonacci polynomials and Lucas polynomials have received less attention than their counterparts for numerical sequences, even if many of these identities can be proved easily. A natural question to ask is: under what conditions is it possible to extend identities that already exist for Fibonacci and Lucas numbers to the GFP? We observe here that the identities involving Fibonacci and Lucas numbers extend naturally to the GFP that satisfy closed formulas similar to the Binet formulas satisfied by Fibonacci and Lucas numbers.

In fact, the well-known sequences below are special cases of the generalized polynomials sequence

• Putting and reduces to Fibonacci polynomials.

• Substituting and yields Lucas polynomials.

• Taking and gives Pell polynomials.

• Taking and gives Pell-Lucas polynomials.

• Taking and gives Jacobsthal polynomials.

• In the case when and it reduces to Jacobsthal-Lucas polynomials.

• In the case when and it reduces to Chebyshev polynomials of first kind.

• In the case when and it reduces to Chebyshev polynomials of second kind.

• Putting and we obtain Chebyshev polynomials of third kind.

• Substituting and yields Chebyshev polynomials of fourth kind.

• Taking and we get Gaussian Pellpolynomials.

• Putting and we obtain Gaussian Jacobsthal polynomials.

• Putting and we obtain Gaussian Jacobsthal-Lucas polynomials.

In order to determine generating functions of generalized polynomials sequence, we use analytical means and series manipulation methods. In the sequel, we derive new symmetric functions and some new properties. We also give some more useful definitions which are used in the subsequent sections. From these definitions, we prove our main results given in Section 3.

2. Definitions and Some Properties

In order to render the work self-contained we give the necessary preliminaries tools; we recall some definitions and results.

Proposition 1. (Favard's Theorem ¹). Let be a monic polynomial sequence. Then is orthogonal if and only if there exist two sequences of complex numbers and such that and satisfies the three-term recurrence relation

(2.1)

Remark 2. If so is called symmetric when

Definition 3. Let and be two positive integers and are set of given variables the k-th elementary symmetric function is defined by

with or .

Definition 4. Let and be two positive integers and are set of given variables the k-th complete homogeneous symmetric function is defined by

with .

Remark 5. Set and by usual convention. For we set and .

Definition 6. ² Let and be any two alphabets. We define by the following form

(2.2)

with the condition for

Equation (2.2) can be rewritten in the following form

(2.3)

where

Definition 7. ³ Given a function on , the divided difference operator is defined as follows

Definition 8. ⁴ the symmetrizing operator is defined by

for all

3. Generating Function of Generalized Polynomials of Second Order

The following theorem is one of the key tools of the proof of our main result which is already given its proof in ⁵.

Theorem 9. Given an alphabet two sequences such that , then

If for the case The following lemmas allow us to obtain many generating functions of generalized polynomials and some well-known polynomials cited above, using a technique symmetric functions.

Lemma 10. Given an alphabet we have

(3.1)

Lemma 11. Given an alphabet we have

(3.2)

• Replacing by in the formulas (3.1) and (3.2), we have

(3.3)

(3.4)

Multiplying the equation (3.3) by and adding it from (3.4) multiplying by and setting we obtain

(3.5)

and we have the following Proposition.

Proposition 12. For the new generating function of generalized polynomials is given by

with

(3.6)

Proof. Theordinary generating function associated is defined by

Using the initial conditions, we get

which is equivalent to

therefore

Accordingly, we conclude the following Corollaries.

Corollary 13. For the generating function of Fibonacci polynomials is given by

(3.7)

Replacing by in the formula (3.7), we have

Corollary 14. For the generating function of Lucas polynomials is given by

(3.8)

Replacing by in the formula (3.8), we have

Corollary 15. For the generating function of Pell polynomials is given by

Corollary 16. For the generating function of Pell-Lucas polynomials is given by

Corollary 17. For the generating function of Jacobsthal polynomials is given by

(3.9)

Replacing by in the formula (3.9), we have

Corollary 18. Forthe generating function of Jacobsthal- Lucas polynomials is given by

(3.10)

Replacing by in the formula (3.10), we have

Corollary 19. For the generating function of Gaussian- Pell polynomials is given by

(3.11)

Replacing by in the formula (3.11), we have

Corollary 20. For the generating function of Gaussian -Jacobsthal polynomials is given by

(3.12)

Replacing by in the formula (3.12), we have

Corollary 21. Forthe generating function of Gaussian –Jacobsthal- Lucas polynomials is given by

(3.13)

Replacing by in the formula (3.13), we have

• Replacing by and by in the formulas (3.1) and (3.2), we have

(3.14)

(3.15)

Multiplying the equation (3.14) by and adding it from (3.15) multiplying by and setting we obtain

(3.16)

and we have the following Proposition.

Proposition 22. Forthe new generating function of generalized polynomials is given by

(3.17)

Corollary 23. Forthe generating function of Chebyshev polynomials of first kind is given by

Corollary 24. For the generating function of Chebyshev polynomials of second kind is given by

Corollary 25. For the generating function of Chebyshev polynomials of third kind is given by

Corollary 26. Forthe generating function of Chebyshev polynomials of fourth kind is given by

4. Conclusion

In this paper, by making use of equations (3.1) and (3.2), we have derived some new generating functions for generalized polynomials of second order. It would be interesting to apply the methods shown in the paper to families of other special polynomials.

Acknowledgments

The authors would like to thank the anonymous referees for their valuable comments and suggestions.

References