# Boubaker Polynomials/Wikipedia/White Fennec/New version 2011

## Boubaker Polynomials

Polynômes de Boubaker Bn(x), avec n de 0 à 6.

According to P. Barry et al.[1], A. Kumar[2] and N. B. Devins et al. (in the Encyclopedia of Physics Research) [3], The Boubaker polynomials are the components of a polynomial sequence [4]:

{\displaystyle {\begin{aligned}B_{0}(x)&{}=1\\B_{1}(x)&{}=x\\B_{2}(x)&{}=x^{2}+2\\B_{3}(x)&{}=x^{3}+x\\B_{4}(x)&{}=x^{4}-2\\B_{5}(x)&{}=x^{5}-x^{3}-3x\\B_{6}(x)&{}=x^{6}-2x^{4}-3x^{2}+2\\B_{7}(x)&{}=x^{7}-3x^{5}-2x^{3}+5x\\B_{8}(x)&{}=x^{8}-4x^{6}+8x^{2}-2\\B_{9}(x)&{}=x^{9}-5x^{7}+3x^{5}+10x^{3}-7x\\&{}\,\,\,\vdots \end{aligned}}}

The Boubaker polynomials are also defined in general mode through the formula:

${\displaystyle B_{n}(x)=\sum _{p=0}^{\lfloor n/2\rfloor }{\frac {n-4p}{n-p}}{\binom {n-p}{p}}(-1)^{p}x^{n-2p}}$

The Boubaker polynomials are also be defined through the differential equation:

{\displaystyle {\begin{aligned}(x^{2}-1)(3nx^{2}+n-2)y{''}+3x(nx^{2}+3n-2)y{'}-n(3n^{2}x^{2}+n^{2}-6n+8)y=0\,\end{aligned}}}

## Integer Sequences generated by the Boubaker polynomials”

The Boubaker polynomials have generated many integer sequences in the On-Line Encyclopedia of Integer Sequences (OEIS)[5]:

## Fields of applications

The Boubaker polynomials have been widely used in different scientific fields:

### Cryogenics

Heat equation inside low-temperatures vessels has been solved using the Boubaker Polynomials Expansion Scheme BPES. The results published by Allyson E. Hayes represent reliable and exploitable temperature profiles between -252°C and -233°C [6] [7].

### Biology and Biophysics

The works of B. Dubey et al. [8] provided analytical solutions to the well-known Lotka-Volterra Predator-Prey equations in the case of quickly satiable predators. The model incorporated an original accelerated-predator-satiety function which is claimed to be closer to reality, and used the Boubaker Polynomials Expansion Scheme BPES. Thanks to this model, it has been demonstrated that, oppositely to most of the predator-pray problem scenarios predation is not strictly proportional to the prey density in presence predators which are ‘never not hungry’.

### Dynamic Systems

A. Milgram used the Boubaker Polynomials Expansion Scheme BPES in order to discuss the stability of some dynamic systems [9].

### Non-Linear Systems

Non-Linear Systems have been also subjected to Comparative Boubaker Polynomials Expansion Scheme (BPES) and Homotopy Perturbation Method (HPM) analysis by H. Koçak et al.[10] and A. Yildirim [11]

### Approximation Theory

Paul Barry and Aoife Hennessy outlined the role of the Boubaker polynomials and their associated integer sequences in arrays analysis and approximation theory [12]

### Thermodynamics and Calorimetry

In the field of thermodynamics and calorimetry, H. Koçak used the Boubaker Polynomials Expansion Scheme BPES in order to determine the coefficients of Antoine vapor-pressure equation coefficients [13], as well as an analytical expression to temperature-dependent Kirkwood-Fröhlich dipole orientation parameter [14]. In the same context, A. Belhadj et al. performed accurate thermal profiles inside Laser keyholes using the same scheme [15][16]

### Mechanics and Hydrology

NASA Astrophysics Data System published a study on Non-Linear Mechanics, carried out by D. H. Zhang , who used the Boubaker Polynomials Expansion Scheme (BPES) for identification of a non-linear 2-degree-of-freedom mechanical system. Thanks to these polynomials, differential equations governing mechanical system behaviours have been transformed into algebraic equations and solutions were plotted in the frequency–energy plane.[17]. Similarly, E. G. Ellouze et al. established the Boubaker Polynomials Copula as a tool for solving hydrological bivariate problems [18]. E. G. Ellouze et al. applied the Boubaker Polynomials copula to a set of discrete random vectors possessing uniform margins. They further suggested a pragmatic way to fit the dependence structure of multivariate data to Boubaker Polynomials copula and empirical contingency tables, and finally established applications of the relationship between infiltration index and the average intensity of rainfall event in some zones.

### Molecular Dynamics

In the field of molecular-scale dynamics, W. X. Yue et al. evaluated water molecule dipole orientation parameters [19] using the Boubaker Polynomials

### Fundamental Mathematics and Fundamental Physics

The Boubaker Polynomials have been used in fundamental mathematics as tools for solving some standard boundary value problems. ) D. H. Zhang et al. [20] proposed an analytical solution to well known applied-physics-related Klein-Gordon equation. Trough some examples, D. H. Zhang et al. presented good fundaments to the Boubaker Polynomials Expansion Scheme BPES , particularly when exact solutions expressions were difficult to establish. In fundamental Physics, works of M. Agida et al. have also used the Boubaker Polynomials Expansion Scheme BPES in order to find exact analytical, piecewise continuous and differentiable solutions to Love’s equation[21],

### Photovoltaics

Boubaker polynomials have been applied in the domain of material characterization. Fridjine et al. Use these polynomials differential equation and algebraic equation in order to investigate Photovoltaic-thermal hybrid solar cells materials[22]

### Algebra, Complex Analysyis,Matrix Analysis and Cryptography

The contribution of the Boubaker polynomials [23] in Pure and applied algebra can be seen through the publications of A. Lzon et al. [24], S. Kumar [25] K. Bülent et al. [26], C. R. Caldera et al. [27] [28] and B. T. Rao et al. [29]

## References

1. Paul Barry and Aoife Hennessy, Journal of Integer Sequences (JIS),Meixner-Type Results for Riordan Arrays and Associated Integer Sequences, Chapter 6: The Boubaker polynomials [1]
2. A. S. Kumar , An analytical solution to applied mathematics-related Love's equation using the ‘’’Boubaker polynomials’’’ expansion scheme| journal=International Journal of the Franklin Institute (elsevier) [2]
3. Encyclopedia of Physics Research, Chapter 21: Cryogenics Vessels Thermal Profilng Using the Boubaker Polynomials, Editors: Nancy B. Devins and Jillian P. Ramos, [3]
4. O.D. Oyodum, O.B. Awojoyogbe, M.K. Dada, J.N. Magnuson, Eur. Phys. J. Appl. Phys. Volume 46, pages 2120-21202, On the earliest definition of the Boubaker polynomials , [4]
5. On-Line Encyclopedia of Integer Sequences
6. citation|title= Book:Cryogenics: Theory, Processes and Applications, Chapter 8: Cryogenics Vessels Thermal Profilng Using the Boubaker Polynomials Expansion Scheme Investigation , Editor: Allyson E.Hayes [5]
7. Satomi Matsumoto and Ueda Iwate, Book:Materials Science Researcher Biographical Sketches and Research Summaries, Cryogenics Vessels Thermal Profilng Using the Boubaker Polynomials Expansion Scheme Investigation,Chapter: Cryogenics Vessels Thermal Profilng Using the Boubaker Polynomials Expansion Scheme Investigation
8. Journal of Theoretical Biology (Elsevier)|id=doi:10.1016/j.jtbi.2010.12.002 B. Dubey, T.G. Zhao, M. Jonsson, H. Rahmanov,A solution to the accelerated-predator-satiety Lotka–Volterra predator–prey problem using Boubaker polynomial expansion scheme, [6]
9. Journal of Theoretical Biology (Elsevier)|id=doi:10.1016/j.jtbi.2010.01.026 A. Milgram|title = The stability of the Boubaker polynomials expansion scheme (BPES)-based solution to Lotka-Volterra problem | [7]
10. Mathematical and Computer Modelling(Elsevier)|iddoi:10.1016/j.mcm.2011.02.031 H. Koçak, A. Yıldırım, D.H. Zhang, S.T. Mohyud-Din,The Comparative Boubaker Polynomials Expansion Scheme (BPES) and Homotopy Perturbation Method (HPM) for solving a standard nonlinear second-order boundary value problem,http://www.citeulike.org/article/8940425
11. The 7th International Conference on Differential Equations and Dynamic Systems, University of South Florida, Tampa, Fmorida USA, 15-18 December 2010 <Page 40 > A. Yildirim,The boubaker polynomials expansion scheme for solving nonlinear science problems, http://web3.cas.usf.edu/main/depts/mth/7thde/data/Abstracts-7thDEDS-Tampa.pdf
12. Journal of Integer Sequences (JIS)Paul Barry, Aoife Hennessy,Meixner-Type Results for Riordan Arrays and Associated Integer Sequences, Chapter 6: The Boubaker polynomials [8]
13. Russian Journal of Physical Chemistry A, Focus on Chemistry (Springer) H. Koçak, Z. Dahong, A. Yildirim,A range-free method to determine antoine vapor-pressure heat transfer-related equation coefficients using the Boubaker polynomials expansion scheme [9]
14. Indian Journal of Physics(Springer) H. Koçak, Z. Dahong, A. Yildirim,Analytical expression to temperature-dependent Kirkwood-Fröhlich dipole orientation parameter using the Boubaker Polynomials Expansion Scheme (BPES) http://www.springerlink.com/content/173787083245t267/
15. Journal of Thermophysics and Heat Transfer (American Institute of Aeronautics and Astronautics) AIAA, A. Belhadj, O. F. Onyango and N. Rozibaeva,Boubaker Polynomials Expansion Scheme-Related Heat Transfer Investigation Inside Keyhole Model [10]
16. Journal of Thermal Analysis and Calorimetry(Akadémiai Kiadó, Springer Science & Kluwer Academic Publishers B.V.), id=doi:10.1007/s10973-009-0094-4, A. Belhadj, J. Bessrour, M. Bouhafs and L. Barrallier,Experimental and theoretical cooling velocity profile inside laser welded metals using keyhole approximation and Boubaker polynomials expansion [11]
17. International Journal of Non-Linear Mechanics (NASA Astrophysics Data System) D. H. Zhang,Study of a non-linear mechanical system using Boubaker polynomials expansion scheme BPES [12]
18. Studies in Nonlinear Sciences (SNS)Emna Gargouri-Ellouze, Noreen Sher Akbar, Sohail Nadeem,Modelling Nonlinear Bivariate Dependence Using the Boubaker Polynomials Copula The Boubaker polynomials [13]
19. Journal of Structural Chemistry (Springer) W. X. Yue, H. Koçak, D. H. Zhang , A. Yıldırım,A second attempt to establish an analytical expression to steam-water dipole orientation parameter using the Boubaker polynomials expansion scheme http://www.springerlink.com/content/57681724u74gvg76/
20. Applied Sciences,(Balkan Society of Geometers, Geometry Balkan Press) D. H. Zhang, L. Naing,The Boubaker polynomials expansion scheme BPES for solving a standard boundary value problem [14]
21. El. Journal of theretical physics ( EJTP), M. Agida , A. S. Kumar, A Boubaker Polynomials Expansion Scheme Solution to Random Love’s Equation in the Case of a Rational Kernel [15]
22. Modern Physics Letters B ([ISSN: 0217-9849, by WS: World Scientific Publishing Co Pte Ltd] ), S. Fridjine and M. Amlouk,A New Parameter-Abacus for optimizing PV-T Hybrid solar devices functional materials using Boubaker Polynomials Expansion Scheme [16]
23. The definition of the Boubaker Poynomials, H. Bannour's Website [17]
24. Recurrence relation for polynomial sequences via Riordan matrices, Pages 24-25: BOUBAKER POLYNOMIALS associated Riordan matrix, A. Luzon and M. Moron [18]
25. International Journal of the Franklin Institute (elsevier), A. S. Kumar , An analytical solution to applied mathematics-related Love's equation using the ‘’’Boubaker polynomials’’’ expansion scheme [19]
26. Kiliç Bülent, Erdal Bas, Page 7, Citation 27: Boubaker polynomials , [20]
27. C. R. Caldera and A. Milgram, , Notes on uniqueness of the Boubaker Polynomials Expansion Scheme (BPES) solution in the case of the Klein–Gordon equation ,Computers and Mathematics with Applications 62 (2011) 536–538, Elsevier [21]
28. C. R. Caldera and A. Milgram, , Boubaker Polynomial Expansion Scheme BPES ternary materials optimization: A critical approach (Comment on a paper of MLBLUE),Material Letters (Elsevier) [22]
29. B. Tirimula Rao, P. Srinivsu, C. Anantha Rao, K. Satya Vivek Vardhan, Jami Vidyadhari ,Page 8 : Boubaker polynomials ,[23]