# Boubaker Polynomials

The Boubaker polynomials are the components of a polynomial sequence [1]:

\begin{align} 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{align}

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

\begin{align} B_0(x) &= 1, \\ B_1(x) &= x, \\ B_2(x) &= x^2+2, \\ B_m(x) &= xB_{m-1}(x) - B_{m-2}(x) \quad\text{for } m>2. \end{align}

Another possible definition of these polynomials is:

$B_n(x)=\sum_{p=0}^{\lfloor n/2\rfloor}\frac{n-4p}{n-p} \binom{n-p}{p} (-1)^p x^{n-2p}$

Alternatively, the Boubaker polynomials can be defined through the differential equation:

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

The Boubaker polynomials have generated many integer sequences in the On-Line Encyclopedia of Integer Sequences (OEIS)[2] and PlanetMath.

## Applications

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

## References

1. 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 , [1]
2. Sequences A135929 , A135936 by Neil J. A. Sloane, A137276 by Roger L. Bagula et Gary Adamson,A138476 , by A. Bannour, A137289, A136256, A136255 by R. L. Bagula à On-Line Encyclopedia of Integer Sequences
3. 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 https://www.novapublishers.com/catalog/product_info.php?products_id=17332&osCsid=06f25d4f739dc8ec36c5160f480acaef
4. 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,http://www.ncbi.nlm.nih.gov/pubmed/20109470
5. Journal of Theoretical Biology (Elsevier)|id=doi:10.1016/j.jtbi.2010.01.026 A. Milgeam|title = The stability of the Boubaker polynomials expansion scheme (BPES)-based solution to Lotka–Volterra problem | http://www.ncbi.nlm.nih.gov/pubmed/21145326
6. 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
7. 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
8. Journal of Integer Sequences (JIS)Paul Barry, Aoife Hennessy,Meixner-Type Results for Riordan Arrays and Associated Integer Sequences, Chapter 6: The Boubaker polynomials http://www.emis.ams.org/journals/JIS/VOL13/Barry5/barry96s.pdf
9. 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 http://www.springerlink.com/content/d78h761823628gl2/
10. 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/
11. Jornal 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| http://pdf.aiaa.org/jaPreview/JTHT/2009/PVJA41850.pdf
12. 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 http://adsabs.harvard.edu/abs/2011IJNLM..46..443Z
13. 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 http://idosi.org/sns/2(1)11/3.pdf
14. 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/
15. 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 http://www.mathem.pub.ro/apps/v12/A12-zh.pdf
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| http://www.springerlink.com/content/2l03064124057686/?p=15de2fa57ce5478aa8a62c2b3a618213&pi=1
17. Heat and Mass Transfer(Springer Berlin / Heidelberg)|id= Volume 45, Number 10 / août 2009, pages:1247-1251 doi:10.1007/s00231-009-0493-x S. Amir Hossein A. E. Tabatabaei, T. Gang Z., O. Bamidele A. and Folorunsho O. Moses,Cut-off cooling velocity profiling inside a keyhole model using the Boubaker polynomials expansion scheme| http://www.citeulike.org/article/4834321 | http://www.springerlink.com/content/b125h6166r216313/
18. 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 OTIMIZING PV-T HYBRID SOLAR DEVICES FUNCTIONAL MATERIALS USING BOUBAKER POLYNOMIALS EXPANSION SCHEME| http://www.worldscinet.com/mplb/23/2317/S0217984909020321.html
19. citation T. G. Zhao, Y. X. Wang and K. B. Ben Mahmoud| title=Limit and uniqueness of the Boubaker-Zhao polynomials single imaginary root sequence | journal=International Journal of Mathematics and Computation | volume=1 |number=08 | ISSN=0974-5718 | http://ceser.res.in/ijmc.html
20. A. Luzon , M. | last2=Moron | | title=RECURRENCE RELATIONS FOR POLYNOMIAL SEQUENCES VIA RIORDAN MATRICES, Pages 24-25: BOUBAKER POLYNOMIALS associated Riordan matrix | http://arxiv.org/PS_cache/arxiv/pdf/0904/0904.2672v1.pdf
21. M. Agida , A. S. . | last2=Kumar |title=A Boubaker Polynomials Expansion Scheme Solution to Random Love’s Equation in the Case of a Rational Kernel || journal=El. Journal of theretical physics ( EJTP) | http://www.ejtp.com/articles/ejtpv7i24p319.pdf
22. 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) | http://cat.inist.fr/?aModele=afficheN&cpsidt=23388093
23. B. Tirimula Rao, P. Srinivsu, C. Anantha Rao, K. Satya Vivek Vardhan , Jami Vidyadhari ,Page 8 : Boubaker polynomials ,http://papers.ssrn.com/sol3/Delivery.cfm/SSRN_ID1523651_code1403499.pdf?abstractid=1523651&mirid=3
24. Kiliç Bülent, Erdal Bas, Page 7, Citation 27: Boubaker polynomials , http://cujse.cankaya.edu.tr/archive/14/02_cujse_10018.pdf