PlanetMath: Boubaker polynomials

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Boubaker polynomials

(Definition)

$\displaystyle B_n(X)=\sum_{p = 0}^{\xi(n)} (-1)^p \frac{(n-4p)}{(n-p)}\binom{n-p}{p}(X)^{n-2p}$

where :

$\displaystyle \xi(n)=\left \lfloor \frac{n}{2} \right \rfloor =\frac{2^n+((-1)^n-1)}{4}$

(The symbol : $\lfloor . \rfloor$ designates the floor function.)

$\displaystyle f_B(X,t)=\frac{1+3t^2}{1+t(t-X)}$

The first few Boubaker polynomials are:

$\displaystyle B_0(X)=1$

$\displaystyle B_1(X)=X$

$\displaystyle B_2(X)=X^2+2$

$\displaystyle B_3(X)=X^3+X$

$\displaystyle B_4(X)=X^4-2$

$\displaystyle B_5(X)=X^5-X^3-3X$

$\displaystyle B_6(X)=X^6-2X^4-3X^2+2$

$\displaystyle B_7(X)=X^7-3X^5-2X^3+5X$

$\displaystyle B_8(X)=X^8-4X^6+8X^2-2$

$\displaystyle B_9(X)=X^9-5X^7+3X^5+10X^3-7X$

$\displaystyle ...$

The Boubaker-Tuki polynomials (or Modified Boubaker polynomials ,noted: or $\tilde {B}_n (X)$ ) are an enhanced form of theBoubaker polynomials. They were named after Pr. Boubaker-Mustapha Rachid TURKI (1934-1987). The Boubaker-Turki polynomials are demonstrated to be solutions of the differential equation:

$\displaystyle (X^2-1)(3nX^2+n-2)y{''}+3X(nX^2+3n-2)y{'}-n(3n^2X^2+n^2-6n+8)y=0$

Bibliography

1: Chaouachi A. et al.
Enhancement of pyrolysis spray disposal performance using thermal time-response to precursor uniform deposition
Eur. Phys. J. Appl. Phys.,37: 105/109.2007.
Link
2: Labiadh H. et al.
A Sturm-Liouville shaped characteristic differential equation as a guide to establish a quasi-polynomial expression to the Boubaker polynomials
Journal of Differential Equations and Control Processes.,37: 105/109.2007.
Available online
3: 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.
Available online
4: Boubaker K.
THE BOUBAKER POLYNOMIALS, A NEW FUNCTION CLASS FOR SOLVING BI-VARIED SECOND ORDER DIFFERENTIAL EQUATIONS.
Journal of Applied Mathematics. F.E.,Volume 31, Issue 3, Pages 273 - 335 (June 2008).
Available online
5: The OnLine Encyclopedia of Integer Sequences.
Sequence number: A138034.
OLEIS,A138034,Boubaker polynomials Bn(X) evaluated at X=1 and X=-1.
Link

"Boubaker polynomials" is owned by liuzuowen. [ full author list (3) | owner history (2) ]

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