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Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
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Biomedical Science and Engineering

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Symmetric Functions of Generalized Polynomials of Second Order

Hind Merzouk, Ali Boussayoud*, Mourad Chelgham

LMAM Laboratory and Department of Mathematics, Mohamed Seddik Ben Yahia University, Jijel, Algeria

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.

Keywords: generating functions, generalized polynomials of second order, symmetric functions

Received September 06, 2019; Revised September 10, 2019; Accepted October 18, 2019

Citation: Hind Merzouk, Ali Boussayoud, Mourad Chelgham, Symmetric Functions of Generalized Polynomials of Second Order. Turkish Journal of Analysis and Number Theory, 7 (5). 135-139. January 2019. doi:10.12691/tjant-7-5-2

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

[picture] | (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

[picture] | (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

[picture] | (2.2)

with the condition for

Equation (2.2) can be rewritten in the following form

[picture] | (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

[picture] | (3.1)

Lemma 11. Given an alphabet we have

[picture] | (3.2)

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

[picture] | (3.3)

[picture] | (3.4)

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

[picture] | (3.5)

and we have the following Proposition.

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

with

[picture] | (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

[picture] | (3.7)

Replacing by in the formula (3.7), we have

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

[picture] | (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

[picture] | (3.9)

Replacing by in the formula (3.9), we have

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

[picture] | (3.10)

Replacing by in the formula (3.10), we have

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

[picture] | (3.11)

Replacing by in the formula (3.11), we have

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

[picture] | (3.12)

Replacing by in the formula (3.12), we have

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

[picture] | (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

[picture] | (3.14)

[picture] | (3.15)

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

[picture] | (3.16)

and we have the following Proposition.

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

[picture] | (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

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