# Boubaker Polynomials

## Definitions and recurrence relations

Boubaker polynomials are the components of a polynomial sequence [1][2]:

{\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}}}

Boubaker polynomials are also defined in general mode through the recurrence relation:

{\displaystyle {\begin{aligned}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{aligned}}}

Note that the first three polynomials are explicitly defined, and that the formula can only be used for m > 2. Another definition of Boubaker polynomials is:

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

Boubaker polynomials can 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}}}

Boubaker polynomials have generated many integer sequences in the w:On-Line Encyclopedia of Integer Sequences [3] and are covered on PlanetMath.

## Controversy

Several times, last time in 2009, Wikipedia chose not to host an article on the subject of Boubakr polynomials, see w:Wikipedia:Articles for deletion/Boubaker polynomials (3rd nomination). This resource is about the polynomials and applications. However, the history of Wikipedia treatment of this topic and users involved with this topic may be studied and discussed on our subpage: /Wikipedia.

" The Boubaker polynomials were established for the first by Boubaker et al. (2006) as a guide for solving a one-dimensional formulation of heat transfer equation...
${\displaystyle {\frac {\partial ^{2}f(x,t)}{\partial x^{2}}}=k{\frac {\partial f}{\partial x}}}$      (on the domain -H<x<0 and t>0) "

This is a direct quote from: Boubaker, K., "On modified Boubaker polynomials: some differential and analytical properties of the new polynomials issued from an attempt for solving bi-varied heat equation," Trends in Applied Sciences Research, 2(6), 540-544. [2]

This comment was appended here: "There is no 2006 reference in this article, and the reference cited as 'accepted' in 2007 cannot be found on Google Scholar."

Students who pay close attention to detail often find errors in peer-reviewed publications, but such errors may also exist in interpretation. The sentence quoted above is in the cited paper by Boubaker. There is, as noted, no 2006 reference in the article, and the article is not footnoted. There are, instead, references:

Boubaker, K., 2007. The Boubaker polynomials, a new function class for solving bi-varied second-order differential equations: F.E.J. Applied Math (Accepted).
Boubaker, K., A. Chaouachi, M. Amlouk, and H. Bouzouita, 2007. Enhancement of pyrolysis spray disposal performance using thermal time-response to precursor uniform deposition. Eur. Phys. J. Applied Phys. 35: 105-109.

The second source first page can be seen at [3]. The publication information given there is

Accepted 12 October 2006.
Published online 26 January 2007.

Since the quoted text refers to Boubaker et al, it is referring to the second reference, not the first. The second reference was accepted in 2006, and since date may have been considered important, the acceptance date was given, or even possibly the submission date. This was simply not made clear.

However, where is the first paper? It is cited in Dada et al, 2009, Establishment of a Chebyshev-dependent Inhomogeneous Second Order Differential Equation for the Applied Physics-related Boubaker-Turki Polynomials, J. Appl. Appl. Math, Vol 3 Issue 2, 329 – 336 [4], this way:

Boubaker K. (2008). The Boubaker polynomials, a new function class for solving bi-varied second order differential equations, F. E. J. of Applied Mathematics, Vol. 31, Issue 3 pp. 273-436.

The paper is also cited in this 2015 "in press" publication: [5] (Boubaker is one of the authors).

The title of the paper is present on Research Gate, with more details, but the actual paper hosted there is the Applied Science paper, not the original one.[6]. This is what is shown as to the original:

The Boubaker polynomials, A new function class for solving bi-varied second order differential equations
Karem Boubaker
Far East Journal of Applied Mathematics 01/2008; 31(3).
ABSTRACT This study presents new polynomials issued from an attempt to solve heat bi-varied equation in a particular case of one-dimensional model. The polynomials, baptized Boubaker polynomials are defined by a recursive formula, which is a critical part of resolution process; they have a demonstrated explicit forms and some interesting properties.

This is the original abstract from the publisher: [7]. It shows a received date of March 14, 2007, but was not published until June, 2008. The acceptance date is not given.

Implications of this research may be covered in analysis to be added to our subpage: /Wikipedia.

The importance of this heat equation in applied mathematics is uncontroversial, as is illustrated in the next section.

## Applications

Boubaker polynomials have been 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, Number 2, May 2009, article number 21201, Comment on “Enhancement of pyrolysis spray disposal performance using thermal time-response to precursor uniform deposition” by K. Boubaker, A. Chaouachi, M. Amlouk and H. Bouzouita. On the earliest definition of the Boubaker polynomials", [pay wall
2. Milovanovic, Gradimir V., et al. "Some properties of Boubaker polynomials and applications." AIP Conference Proceedings-American Institute of Physics. Vol. 1479. No. 1. 2012. http://www.mi.sanu.ac.rs/~gvm/radovi/ASCA-1050-1053.pdf
3. oeis.org Sequences A135929, A135936 by Neil J. A. Sloane, A137276 by Roger L. Bagula and Gary Adamson, A138476 , by A. Bannour, A137289, A136256, A136255, by R. L. Bagula, A160242 by A. Rahmanov,
4. 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
5. 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
6. 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
7. 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
8. 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
9. 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
10. 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/
11. 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/
12. 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
13. D.H. Zhang, "Study of a non-linear mechanical system using Boubaker polynomials expansion scheme BPES," International Journal of Non-Linear Mechanics Volume 46, Issue 2, March 2011, Pages 443–445.[1]
14. 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
15. 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/
16. 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
17. 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
18. 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/
19. 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
20. 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
21. 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
22. 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
23. 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
24. 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
25. Kiliç Bülent, Erdal Bas, Page 7, Citation 27: Boubaker polynomials , http://cujse.cankaya.edu.tr/archive/14/02_cujse_10018.pdf

## Sources

Subpage for the collection of sources on Boubaker polynomials: /Sources

## Resources

• /Wikipedia study of Wikipedia history re this topic
• /Sources list of peer-reviewed sources categorized as having Boubaker as author or co-author