# Boubaker Polynomials

## Definitions and recurrence relations

Boubaker polynomials are the components of a polynomial sequence :

{\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:

{\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:

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

Boubaker polynomials can be defined through the differential equation:

{\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  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...
${\frac {\partial ^{2}f(x,t)}{\partial x^{2}}}=k{\frac {\partial f}{\partial x}}$       (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. 

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 . The publication information given there is

Received 9 May 2006.
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 , 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:  (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.. 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: . 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: